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 (xn) doesn't converge uniformly on the closed intervall (0,1)