Sum of two sets of functions Let L,K in $C[0,1]$ be compact. Prove or give a counter example that $L+k$ is compact. 
Here $C[0,1]$ is the space of all continuous functions on $[0,1]$ with the sup metric. I think it is compact and my strategy is to show every sequence has a converging subsequence. Let $x_n=f_n+g_n$ where $f_n$ is in L and $g_n$ is in k. Then both have a converging subsequence so $x_n$ has a converging subsequence since the uniform limit of the sum is the sum of the uniform limits. Is this reasoning correct?

Let $f_n + g_n$ be any sequence in $L+K$, where $\{ f_n \}_n \subset L$ and $\{ g_n \}_n \subset K$.
By compactness of $L$ there exists a subsequence $\{ f_{n_k} \}_k$ converging in $L$.
By compactness of $K$ there exists a subsequence $\{ g_{n_{k_l}}\}_l$ converging in $K$.
Since $\{ f_{n_{k_l}} \}_l$ is a subsequence of a converging sequence, it is convergent.
Thus, the subsequence $\{ f_{n_{k_l}} +g_{n_{k_l}} \}_l$ is convergent in $L+K$.
