Definition:Uniform Convergence

Real Numbers
The above definition can be applied directly to the real numbers treated as a metric space:

Also defined as
Some sources insist that $N \in \N$ but this is unnecessary and makes proofs more cumbersome.

Also see

 * Definition:Convergent Sequence

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.