I am looking at metric spaces. For the metric space C(X,Y) the metric is:
$\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$
This is the proof I am reading about.
I would like to do the proof without just having to write ${lim}_{m \rightarrow \infty}$. That is I would like to take care of m, as the definition of a limit, not just say that it goes to infitity.
Can I do that by writing:
$d_{Y}(f_{n}(x),f(x)) \le d_{Y}(f_{n}(x),f_{m}(x))+d_{Y}(f_{m}(x),f(x)) < \epsilon/2+\epsilon/2=\epsilon$
The problem that occurs here is that we need to show that $f_{n}(x)$ converges uniformely to f(x). So for a given $\epsilon$ we must find an N such that if $n \ge N$ then $d_{Y}(f_{n}(x),f(x)) < \epsilon $. But in doing this proof the problem is that I only know $f_{m}(x)$ converges pointwise. But is still this not a problem ? Because the definition says I only need to find an N, an even though for each x there may be a new m, and even though we may never find an m so that this holds for all x this does not matter?, because we do not require m to be a finite number?
Basically, i can not find an m where the equation holds? But does that create trouble?