When can you interchange limits of a sequence of 2 variables?

Question

$$\lim_{n\to\infty} \lim_{N\to\infty} a(N,n)$$

the reason I want to know is

I have a sequence of functions $\{f_n\}$ which converges uniformly to a function $f$

and I want to say that

$$\lim_{n\to\infty} \lim_{N\to\infty} \sum_{k=1}^N \frac{f_k(x_n)}N = \lim_{N\to\infty} \lim_{n\to\infty} \sum_{k=1}^N \frac{f_k(x_n)}N $$

here my $a(N,n)$ is that sum ($\sum_{k=1}^N f_k(x_n)/N$)

and somehow I want to use that $f_n$ converges uniformly to do this

sorry.edited

Answer 1

If the total sum converges absolutely, you can reorder the terms however you want. Can you prove that?

Answer 2

If they commute with each other we can interchange the limits........

Answer 3

Is the limit of the $f_j$'s continuous? If so I think your statement is true. Let $g_N(x) = 1/N \ \sum f_j(x)$. Now if $f_j \to f$, then $g_N \to f$ as well (if each $f_j$ is within an epsilon of $f$, then $g_N$ is within an epsilon of $f$ as well; go far enough out in the sequence to make this happen).

Now say $x_n \to x$. Then the RHS is $$ \lim_{N \to \infty} g_N (x) = f(x)$$, and the LHS is $$ \lim_{n \to \infty} f(x_n) = f(x)$$ if $f$ is continuous.