_{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)