Started this problem but can't finish it: Showing pointwise convergence for this summation I know how to start this problem but am having trouble finishing the end of it. Any help would be great! Thanks 
We let $g_n: E \rightarrow \mathbb{R}$ be continuous functions for $1 \leq n \leq N$ and we let $a_k^{(n)}$ be $N$ convergent sequences of numbers. Assume lim$_{k\rightarrow \infty}a_k^{(n)} = a_n$. Let $f = \sum_{n=1}^N a_n g_n$.
I want to show that:
$\sum_{n=1}^{N} a_k^{\{n\}} g_n$ converges point-wise to $f$. 
My solution so far:
Pointwise convergence definition: A sequence of functions $g_n$ defined on a set M converges pointwise to a function g on M if lim $f_n(x)$ = $f(x)$ as n $\rightarrow$ ∞ holds for all $x$ in M.   
I can fix $x$ in $E$ and then look at |$\sum_{n=1}^{N} a_k^{\{n\}}g_n(x)$ - $\sum_{n=1}^{N} a_ng_n(x)$|but from here I am do not know how to break this down to show pointwise convergence. 

Hint: |$\sum_{n=1}^{N} a_k^{\{n\}}g_n(x)$ - $\sum_{n=1}^{N} a_ng_n(x)$| = $|\sum_{n=1}^N (a_k^{\{n\}} - a_n) g_n(x)|$. 
Additionally, $|\sum_{n=0}^N x_n| \leq \sum_{n=0}^N |x_n|$ for any $x_n$ (this is the triangle inequality).
Can you take it from here?
