Definition:Uniform Convergence
Extend this definition to whatever other topological spaces this can be made to make sense for.
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Definition
Let $S$ be a set.
Let $\left({M, d}\right)$ be a metric space.
Let $\left \langle {f_n} \right \rangle$ be a sequence of functions $f_n : S \to M$.
Suppose that $\forall \epsilon > 0: \exists N \in \R: \forall n \ge N, \forall x \in S: \left\vert{f_n \left({x}\right) - f \left({x}\right)}\right\vert < \epsilon$.
Then $\left \langle {f_n} \right \rangle$ converges to $f$ uniformly on $S$ as $n \to \infty$.
Note
Some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.
Also see
Comment
Note that this definition of convergence of a function is stronger than that for pointwise convergence, in which it is necessary to specify a value of $N$ given $\epsilon$ for each individual point.
In uniform convergence, given $\epsilon$ you need to specify a value of $N$ which holds for all points in the domain of the function.
