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Uniform Convergence — Definition, Formula & Examples

Uniform convergence is a type of convergence for a sequence of functions where the speed of convergence does not depend on which point in the domain you choose. Unlike pointwise convergence, the entire function sequence gets uniformly close to the limit function everywhere at once.

A sequence of functions {fn}\{f_n\} converges uniformly to ff on a set SS if for every ε>0\varepsilon > 0, there exists an N∈NN \in \mathbb{N} (depending only on ε\varepsilon, not on xx) such that for all n≥Nn \geq N and all x∈Sx \in S, ∣fn(x)−f(x)∣<ε|f_n(x) - f(x)| < \varepsilon.

Key Formula

sup⁡x∈S∣fn(x)−f(x)∣→0as n→∞\sup_{x \in S} |f_n(x) - f(x)| \to 0 \quad \text{as } n \to \infty
Where:
  • fnf_n = The nth function in the sequence
  • ff = The limit function
  • SS = The domain on which convergence is tested
  • sup⁡\sup = The supremum (least upper bound) over all x in S

How It Works

To test for uniform convergence, compute Mn=sup⁡x∈S∣fn(x)−f(x)∣M_n = \sup_{x \in S} |f_n(x) - f(x)|. If Mn→0M_n \to 0 as n→∞n \to \infty, the convergence is uniform. For series ∑gn(x)\sum g_n(x), the Weierstrass M-test is a powerful tool: if ∣gn(x)∣≤Mn|g_n(x)| \leq M_n for all x∈Sx \in S and ∑Mn\sum M_n converges, then ∑gn(x)\sum g_n(x) converges uniformly on SS. Uniform convergence guarantees you can swap limits with integrals, and under additional conditions, with derivatives.

Worked Example

Problem: Determine whether fn(x)=xnf_n(x) = x^n converges uniformly on [0,1)[0, 1).
Find the pointwise limit: For each x∈[0,1)x \in [0,1), as n→∞n \to \infty, xn→0x^n \to 0. So the pointwise limit is f(x)=0f(x) = 0.
f(x)=lim⁡n→∞xn=0for x∈[0,1)f(x) = \lim_{n \to \infty} x^n = 0 \quad \text{for } x \in [0,1)
Compute the supremum of the error: The difference is ∣fn(x)−f(x)∣=xn|f_n(x) - f(x)| = x^n. Over the interval [0,1)[0,1), the supremum of xnx^n approaches 1 (as x→1−x \to 1^-) for every nn.
Mn=sup⁡x∈[0,1)xn=1M_n = \sup_{x \in [0,1)} x^n = 1
Conclude: Since Mn=1M_n = 1 does not tend to 0, the convergence is not uniform on [0,1)[0,1).
Mn=1↛0M_n = 1 \not\to 0
Answer: The sequence fn(x)=xnf_n(x) = x^n converges pointwise but not uniformly on [0,1)[0, 1).

Why It Matters

Uniform convergence is essential in real analysis because it lets you interchange limits with integration and, under suitable hypotheses, with differentiation. Power series converge uniformly on compact subsets of their interval of convergence, which is why you can differentiate and integrate them term by term — a fact used constantly in differential equations, physics, and engineering.

Common Mistakes

Mistake: Assuming pointwise convergence is sufficient to swap limits with integrals or to preserve continuity.
Correction: Pointwise convergence alone does not guarantee these properties. You need uniform convergence. A classic counterexample is fn(x)=xnf_n(x) = x^n on [0,1][0,1], where each fnf_n is continuous but the pointwise limit is discontinuous at x=1x=1.