Let be X,Y

metric spaces, d

_{X},d

_{Y} the

metrics of X and Y. Let (f

_{n}) be a

sequence of

functions f

_{n}: X -> Y.

(f

_{n}) is called uniformly converging to a function f: X -> Y

iff
for any

*u* > 0,

*u* of

**R** there exists a

*n* > 0,

*n* of

**N** with:
for all x of X and all

*m* >

*n* d

_{Y}( f

_{m}(x) , f(x) ) <

*u*
Examples:
The sequence of functions (1/n * cos(x) ) converges uniformly to the constant function f(x)=0.

The sequence of functions (x^{n}) doesn't converge uniformly on the closed intervall (0,1)