Definition
Suppose S is a set and fn : S → R is a real-valued function for every natural number n. We say that the sequence (fn)n∈N is uniformly convergent with limit f : S → R if for every ε > 0, there exists a natural number N such that for all x ∈ S and all n ≥ N we have |fn(x) − f(x)| < ε.
Consider the sequence αn = supx |fn(x) − f(x)| where the supremum is taken over all x ∈ S. Clearly fn converges to f uniformly if and only if αn tends to 0.
The sequence (fn)n∈N is said to be locally uniformly convergent with limit f if for every x in some metric space S, there exists an r > 0 such that (fn) converges uniformly on B(x,r) ∩ S.
Read more about this topic: Uniform Convergence
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