Can you me proving , uniform convergence implies pointwise convergence

[shehpar]

I don't know, I started from the definition of uniform convergence and it seems pretty obvious to me , can anybody start me at least towards right direction?

 

[HallsofIvy]

Yes, that pretty much all it takes.

Definition of "uniformly convergent": $\{f_n(x)\}$ converges to f(x) as n goes to infinity uniformly if and only if, given $\epsilon> 0$, there exist N such that if n > N, then $|f_n(x)- f(x)|< \epsilon$.

Definition of "convergent": ${f_n(x)}$ converges to f(a) for given a as n goes to infinity if and only if given $\epsilon> 0$, there exsit N such that if n> N, then $|f_n(a)- f(a)|< \epsilon$.

Basically, "uniformly convergent" requires that, given $\epsilon$, you be able to use the same $\delta$ for every value of x. Just "convergent" means the value of $\delta$ may depend upon $\epsilon$ and the value of x at which the function is evaluated.