Sequences and Function Convergence

The Central Idea

A sequence records an ordered process:

We say the sequence converges when its terms can be forced arbitrarily close to one fixed number by going far enough along the sequence.

For a sequence of functions,

the same basic question becomes richer. We must decide whether the functions approach a limiting function one input at a time or whether a single stage of the sequence works uniformly across the whole domain.

From Informal Approach to Rigorous Convergence

A graph or table can suggest that a sequence approaches a value, but rigorous analysis asks for a guarantee that survives every requested level of accuracy.

The central pattern is:

The word eventually is important. A finite number of early terms may behave irregularly without affecting the limit.

Sequences as Functions on the Natural Numbers

Formally, a real sequence is a function

We usually write the value at the index \(n\) as \(a_n\) rather than \(a(n)\).

Thus the notation

describes the ordered list

Examples of Sequences

The formula

generates

The formula

generates

The first sequence settles toward a single value. The second continues to oscillate between two separated values.

The Definition of Sequence Convergence

A real sequence \((a_n)\) converges to \(L\in\mathbb{R}\) if for every \(\varepsilon>0\), there exists an index \(N\in\mathbb{N}\) such that

We write

or

Reading the Epsilon Definition

The quantity \(\varepsilon\) represents an arbitrary positive tolerance. The definition says that no matter how small that tolerance is, we can find a stage after which every term satisfies

Example: Proving \(1/n\to0\)

Let

We want

This is equivalent to

or

By the Archimedean property, choose an integer \(N>1/\varepsilon\). Then every \(n\ge N\) satisfies

Therefore,

Example: A Shifted Reciprocal Sequence

Consider

We expect \(a_n\to5\). Indeed,

To make this less than \(\varepsilon\), it is enough to require

Hence

Only the Tail Determines the Limit

Suppose a sequence is changed in only finitely many positions. Its long-term tail remains unchanged, so its convergence behavior remains unchanged.

For example,

still converges to \(0\) if its terms agree with \(1/n\) from some point onward.

Limits describe eventual behavior rather than the appearance of the first few terms.

A Convergent Sequence Has Only One Limit

A real sequence cannot converge to two different real numbers.

Suppose \(a_n\to L\) and \(a_n\to M\). If \(L\ne M\), choose

Eventually \(a_n\) must lie within \(\varepsilon\) of both \(L\) and \(M\). The triangle inequality would then give

a contradiction. Therefore,

Every Convergent Sequence Is Bounded

Suppose \(a_n\to L\). Choose \(\varepsilon=1\). Then there is an index \(N\) such that

Thus every tail term lies between \(L-1\) and \(L+1\). The finitely many earlier terms are also bounded as a finite set.

Consequently,

The converse is false: a bounded sequence need not converge.

Bounded Does Not Mean Convergent

The sequence

is bounded because

But it does not converge. Its terms continue alternating between \(-1\) and \(1\), so they never remain inside an arbitrarily small neighborhood of one number.

Divergence

A sequence diverges if it does not converge to any finite real number.

Divergence can occur in several ways. A sequence may:

• oscillate, as \((-1)^n\) does;
• grow without bound, as \(n\) does;
• decrease without bound, as \(-n\) does;
• display more complicated nonconvergent behavior.

These behaviors should not all be treated as the same geometric picture, even though none gives a finite real limit.

Infinite Limits

We write

if for every real number \(M\), there exists \(N\) such that

Similarly,

if eventually \(a_n

The symbols \(\infty\) and \(-\infty\) are not real numbers. These statements describe unbounded behavior rather than convergence to an element of \(\mathbb{R}\).

Limit Laws for Sequences

If

then the familiar algebraic operations preserve limits:

and, if \(B\ne0\) and \(b_n\ne0\) eventually,

Constant Multiples and Powers

If \(a_n\to A\) and \(c\in\mathbb{R}\), then

For every positive integer \(k\),

More generally, continuous functions interact naturally with convergent sequences: if \(a_n\to A\) and \(g\) is continuous at \(A\), then

Example: Using Limit Laws

Evaluate

Divide numerator and denominator by \(n\):

Since

the limit laws give

The Squeeze Principle for Sequences

Suppose

eventually, and suppose

Then

The middle sequence cannot escape a shrinking region whose two boundaries converge to the same point.

Example: A Squeezed Oscillating Sequence

Consider

Since

we have

Both outer sequences converge to \(0\), so

Monotone Sequences

A sequence is increasing if

for every relevant \(n\), and decreasing if

Either type is called monotone.

The Monotone Convergence Theorem

Completeness of the real numbers gives a powerful criterion:

Likewise,

For an increasing bounded sequence,

Example: A Monotone Bounded Sequence

Let

The terms increase because \(1/n\) decreases, and they are bounded above by \(1\). Therefore the monotone convergence theorem guarantees convergence.

In fact,

Subsequences

A subsequence is formed by selecting terms from a sequence while preserving their original order.

If

Convergence Passes to Subsequences

If

then every subsequence also converges to \(L\):

A subsequence eventually uses only terms from the tail of the original sequence, so it inherits the same limiting behavior.

Using Subsequences to Prove Divergence

The sequence

has the even subsequence

and the odd subsequence

These subsequences converge to different limits. Therefore the original sequence cannot converge.

Bolzano–Weierstrass for Sequences

Every bounded real sequence has at least one convergent subsequence.

This does not say that the original sequence converges. The sequence \((-1)^n\) is bounded and divergent, but it has convergent subsequences with limits \(1\) and \(-1\).

Cauchy Sequences

Sometimes we can tell that sequence terms are clustering together before we know the limit.

A sequence \((a_n)\) is a Cauchy sequence if for every \(\varepsilon>0\), there exists \(N\) such that

The definition compares late terms with one another rather than comparing them with a proposed limit.

Convergent Sequences Are Cauchy

Suppose \(a_n\to L\). For sufficiently large \(m\) and \(n\), both terms lie close to \(L\). By the triangle inequality,

If each term on the right is less than \(\varepsilon/2\), then

Therefore every convergent real sequence is Cauchy.

Completeness and the Cauchy Criterion

The completeness of \(\mathbb{R}\) gives the converse:

This equivalence is extremely useful because it lets us establish existence of a limit without first identifying the limit explicitly.

From Number Sequences to Function Sequences

Now suppose every term of a sequence is itself a function defined on a common domain \(D\):

We want to understand what it means for

to approach a limiting function \(f\).

There is more than one useful notion of convergence.

Pointwise Convergence

A sequence of functions \((f_n)\) converges pointwise to \(f\) on \(D\) if for every fixed \(x\in D\),

In epsilon language, for every \(x\in D\) and every \(\varepsilon>0\), there exists an index \(N\) such that

The key feature is that \(N\) may depend on both \(x\) and \(\varepsilon\).

Pointwise Convergence Means One Input at a Time

To test pointwise convergence, freeze an input \(x\). The values

form an ordinary numerical sequence. Find its limit. Then repeat the idea conceptually for every input in the domain.

The resulting value as a function of \(x\) is the pointwise limit.

Example: Pointwise Convergence of \(x/n\)

Define

on \(\mathbb{R}\). For each fixed real number \(x\),

Therefore

Example: Pointwise Limit of \(x^n\) on \([0,1]\)

Let

If \(0\le x<1\), then

At the endpoint \(x=1\),

Hence the pointwise limit is

A Warning from the \(x^n\) Example

Every function

is continuous on \([0,1]\), but the pointwise limit

is discontinuous at \(x=1\).

Therefore pointwise convergence alone does not preserve continuity.

Uniform Convergence

A sequence of functions \((f_n)\) converges uniformly to \(f\) on \(D\) if for every \(\varepsilon>0\), there exists an index \(N\) such that

for every \(x\in D\).

The same \(N\) must work simultaneously for the entire domain.

The Order of Quantifiers Matters

Pointwise convergence says

Uniform convergence says

In the uniform statement, \(N\) is chosen before \(x\), so it cannot depend on the input.

Uniform Convergence Implies Pointwise Convergence

If one index \(N\) works for every input in the domain, then it certainly works for each particular fixed input.

Therefore,

The converse is not generally true.

The Supremum-Norm Test

When the quantity is finite, define the maximum-error measure

Then uniform convergence can be recognized by

This viewpoint turns function convergence into ordinary convergence of a sequence of nonnegative numbers measuring worst-case error.

Example: Uniform Convergence on a Bounded Interval

Let

on \([-5,5]\). The proposed limit is \(f(x)=0\). For every \(x\in[-5,5]\),

Therefore,

Hence

The Same Sequence Need Not Be Uniform on a Larger Domain

Consider again

now on all of \(\mathbb{R}\).

Pointwise convergence to \(0\) still holds because every fixed \(x\) is divided by larger and larger \(n\).

But for any fixed \(n\), the error

is unbounded as \(|x|\to\infty\). No single \(N\) can force the error below a positive tolerance for every real \(x\).

Thus the convergence is not uniform on \(\mathbb{R}\).

Why \(x^n\) Is Not Uniform on \([0,1]\)

The pointwise limit of \(x^n\) on \([0,1]\) is discontinuous at \(1\). Since each \(x^n\) is continuous and a uniform limit of continuous functions would be continuous, uniform convergence is impossible.

We can also see the failure directly. For inputs just below \(1\), the value \(x^n\) can remain close to \(1\) even when \(n\) is large. The convergence slows down near the endpoint.

Uniform Convergence on a Smaller Interval

Fix a number \(r\) with

On \([0,r]\),

Since \(r^n\to0\),

Therefore

Uniform Limits Preserve Continuity

A central theorem states:

The proof works by splitting the difference

into errors involving one sufficiently late function \(f_N\), whose continuity is already known.

The Three-Term Continuity Estimate

Insert and subtract \(f_N(x)\) and \(f_N(a)\):

Uniform convergence controls the first and third terms simultaneously across the domain. Continuity of \(f_N\) controls the middle term near \(a\).

This estimate explains why uniform convergence is strong enough to transfer continuity to the limit.

Uniform Convergence and Integration

Suppose each \(f_n\) is Riemann integrable on a closed bounded interval \([a,b]\) and \(f_n\to f\) uniformly there. Then \(f\) is Riemann integrable and

The key estimate is

Pointwise Convergence Is Not Enough for Interchanging Limits and Integrals

Pointwise convergence controls each input separately, but an integral combines information across the entire domain. A sequence can develop narrow regions where the functions remain large even though each fixed input eventually sees small values.

Later measure-theoretic convergence theorems provide more flexible conditions for passing limits through integrals. For now, uniform convergence gives a clean sufficient condition on closed bounded intervals.

Differentiation Requires More Care

Uniform convergence of differentiable functions does not by itself guarantee that the limiting function is differentiable, nor does it automatically justify

A standard theorem requires stronger hypotheses, such as convergence of the derivatives uniformly together with convergence of the original functions at at least one point on the interval.

This distinction is one reason different convergence concepts matter in analysis.

Function Sequences Can Converge at Different Speeds

In pointwise convergence, the index required for a desired accuracy may vary substantially with \(x\).

Uniform convergence imposes a worst-case requirement: once \(n\ge N\), every point must satisfy the same error tolerance.

This is why uniform convergence is often interpreted as convergence at a common rate across the domain, although the precise statement is about a common index for each tolerance rather than a numerical rate formula.

The Uniform Cauchy Criterion

A sequence of real-valued functions is uniformly Cauchy on \(D\) if for every \(\varepsilon>0\), there exists \(N\) such that

for every \(x\in D\).

Because \(\mathbb{R}\) is complete, a uniformly Cauchy sequence of real-valued functions has a pointwise-defined real limit, and the convergence to that limit is uniform.

Function Spaces Begin Here

Once functions themselves are treated as objects, we can measure distances between them. On a domain \(D\), the quantity

acts like a distance-generating norm whenever it is finite on the function class under consideration.

Uniform convergence then becomes ordinary convergence with respect to this norm:

This viewpoint leads naturally to metric spaces, normed spaces, and functional analysis.

Pointwise and Uniform Convergence Compared

The two ideas can be summarized as follows.

• Pointwise: fix \(x\), then go far enough in \(n\).
• Uniform: go far enough in \(n\) so that every \(x\) works at once.
• Uniform convergence always implies pointwise convergence.
• Pointwise convergence may fail to preserve continuity.
• Uniform convergence preserves continuity.
• On a closed bounded interval, uniform convergence supports interchange of limit and Riemann integral.

A Reliable Procedure for Numerical Sequence Limits

1. Identify the proposed limiting value.
2. Simplify \(|a_n-L|\).
3. Set the error expression less than \(\varepsilon\).
4. Solve for a sufficient condition on \(n\).
5. Choose an integer \(N\) satisfying that condition.
6. State that every \(n\ge N\) meets the desired tolerance.
7. Conclude \(a_n\to L\).

A Reliable Procedure for Pointwise Limits

1. Fix an arbitrary input \(x\) in the domain.
2. Treat \(f_n(x)\) as a numerical sequence in \(n\).
3. Compute or prove its limit.
4. Repeat the description across the domain, noting exceptional points if necessary.
5. Assemble the resulting values into the limiting function \(f(x)\).

A Reliable Procedure for Uniform Convergence

1. Find the pointwise limit \(f\) first.
2. Form the error \(|f_n(x)-f(x)|\).
3. Try to bound that error by a quantity depending only on \(n\).
4. Equivalently, compute or estimate \(\sup_{x\in D}|f_n(x)-f(x)|\).
5. Show that the worst-case error tends to zero.
6. Choose a single \(N\) that works for every \(x\in D\).

How to Show Convergence Is Not Uniform

To disprove uniform convergence, it is enough to find one positive tolerance \(\varepsilon_0\) for which no common index works.

A practical method is to construct inputs \(x_n\) depending on \(n\) such that

for arbitrarily large \(n\).

The input is allowed to vary with \(n\) because uniform convergence must control every input at each sufficiently late stage.

Common Error: Checking Only a Few Terms

Numerical evidence can suggest a limit but cannot prove the epsilon condition. Convergence is a statement about every sufficiently late term, not merely about a large sample of computed values.

Common Error: Letting \(N\) Depend on \(n\)

In the definition of sequence convergence, once \(\varepsilon\) is fixed, we choose one threshold \(N\). The implication must then hold for every \(n\ge N\).

Choosing a new threshold after seeing each \(n\) does not establish the definition.

Common Error: Assuming Bounded Means Convergent

Boundedness is necessary for convergence but not sufficient.

The sequence

is bounded and divergent. Additional structure, such as monotonicity, can turn boundedness into a convergence criterion.

Common Error: Confusing a Sequence with Its Range

A sequence remembers order and repetition. The set of values taken by a sequence does not.

For example, the sequences

and

can use the same values \(\{0,1\}\) while having very different tail behavior. Convergence depends on the ordered process, not only on the range.

Common Error: Proving Pointwise Convergence and Calling It Uniform

In a pointwise proof, the threshold may depend on the selected input:

In a uniform proof, the threshold must have the form

and must work for every input in the domain.

Common Error: Ignoring the Domain

Uniform convergence can depend strongly on the domain.

The sequence

converges uniformly to \(0\) on every bounded interval but not on all of \(\mathbb{R}\).

Always state the domain when discussing function convergence.

Common Error: Assuming Limits Commute with Every Operation

Algebraic operations interact well with ordinary convergent sequences, but operations on functions can require stronger hypotheses.

Passing a limit through an integral or derivative is not automatic. The convergence mode and the surrounding assumptions must be checked.

Example: A Uniform Error Bound

Let

on \([-2,2]\). The pointwise limit is \(0\), and

Since \(4/n\to0\),

Example: A Sequence with a Piecewise Pointwise Limit

Define

on \([0,1]\).

At \(x=0\),

For every fixed \(x>0\), divide numerator and denominator by \(n\):

Therefore the pointwise limit is

Example: Uniform Convergence by Exact Supremum

Let

on \([0,1]\). For each \(x\), the limit is \(0\). Moreover,

The function is increasing in \(x\), so the largest error occurs at \(x=1\):

Therefore,

Convergence as Error Control

Both numerical and function convergence can be viewed through error.

For a number sequence, the error is

For uniform function convergence, the worst-case error is

Analysis turns the informal idea of “getting close” into explicit control of these errors.

Convergence and Approximation

A convergent sequence often supplies a practical approximation scheme. Once an error bound is known, we can choose an index large enough to meet a requested accuracy.

This same principle appears in numerical analysis, Fourier approximation, power series, iterative algorithms, differential equations, and statistical computation.

Practice 1: Epsilon Control

Prove that

converges to \(0\) by finding a sufficient \(N\) in terms of \(\varepsilon\).

Practice 2: Limit Laws

Evaluate

Practice 3: Bounded but Divergent

Explain why

is bounded but does not converge.

Practice 4: Monotone Convergence

Show that

is increasing and bounded above, and identify its limit.

Practice 5: Pointwise Limit

Find the pointwise limit on \([0,1]\) of

Practice 6: Uniform Convergence

Determine whether

converges uniformly to \(0\) on \([-3,3]\).

Practice 7: Domain Dependence

Explain why

converges pointwise but not uniformly on \(\mathbb{R}\).

Practice 8: Pointwise but Not Uniform

Let

on \([0,1]\). State the pointwise limit and explain why the convergence is not uniform.

Practice 9: Subsequences

The sequence

has an even-index and an odd-index subsequence. Find the limits of those two subsequences and use them to classify the original sequence.

Practice 10: Uniform Error

Let

on \([0,2]\). Find an upper bound for

and determine whether \(f_n\to0\) uniformly.

Practice Answers

Practice 1

We need

It is enough to require

Choose any integer

Then every \(n\ge N\) satisfies \(7/n<\varepsilon\), so

Practice 2

Divide by \(n\):

Since \(1/n\to0\),

Practice 3

Since

every term has absolute value \(1\), so the sequence is bounded. Its even subsequence equals \(1\), while its odd subsequence equals \(-1\). Since these subsequences have different limits, the full sequence diverges.

Practice 4

Because \(1/n\) decreases, \(2-1/n\) increases. Also,

so \(2\) is an upper bound. Hence the sequence converges, and directly

Practice 5

For every fixed \(x\in[0,1]\),

Thus the pointwise limit is

Practice 6

On \([-3,3]\),

Therefore,

Hence the convergence is uniform.

Practice 7

For every fixed \(x\in\mathbb{R}\), \(x/n\to0\), so pointwise convergence holds. But for any fixed \(n\),

Therefore no single index controls the error across all real inputs, and the convergence is not uniform.

Practice 8

The pointwise limit is

Every \(f_n(x)=x^n\) is continuous, but the pointwise limit is discontinuous at \(1\). A uniform limit of continuous functions would be continuous, so the convergence cannot be uniform.

Practice 9

For even indices,

For odd indices,

Since two subsequences have different limits, the original sequence diverges.

Practice 10

On \([0,2]\), the function \(x/(n+x)\) increases with \(x\), so

Since

we conclude

What to Remember

Numerical convergence is controlled by the condition

For functions, the essential distinction is whether the stage at which the approximation becomes accurate may depend on the input.

• A convergent real sequence has a unique limit and is bounded.
• A bounded sequence need not converge.
• Every subsequence of a convergent sequence has the same limit.
• In \(\mathbb{R}\), Cauchy sequences and convergent sequences are the same class.
• Pointwise convergence treats each input separately.
• Uniform convergence uses one threshold for the entire domain.
• Uniform convergence implies pointwise convergence, but not conversely.
• Uniform limits of continuous functions are continuous.
• Uniform convergence on a closed bounded interval permits passage of the limit through the Riemann integral.

Where We Go Next

Sequences provide the basic language of approximation, and function convergence shows that mathematical objects can themselves form sequences. The next step is to study spaces in which distance between general objects replaces ordinary distance between real numbers.

The next lesson, Metric Spaces, abstracts the idea of distance and develops convergence, open sets, closed sets, continuity, and completeness in a setting broad enough to include Euclidean spaces and spaces of functions.