Uniform and Pointwise Convergance of Functions

[logarithmic]

Can anyone explain to me why if a sequence of functions {f_n} is uniformly convergent, then it must converge to it's pointwise limit?

So, if we showed {f_n} is not uniformly convergent to some f (f is the pointwise limit), how do we know that there isn't some other function, say g, such that {f_n} converges to g uniformly?

 

[CompuChip]

The reason is that uniform convergence is stronger than pointwise convergence.

Suppose that the limit is f.
In the definition, you first pick some ε > 0.
Roughly speaking, for pointwise convergence you can find for any x some number N = N(x) such that for n > N, |f_n(x) - f(x)| < ε.
For uniform convergence you can find an N' which works for all x at the same time. So if you have uniform convergence and you have your ε you can find some N', and if you take N(x) = N' for all x you can prove pointwise convergence.