Prove that \begin{equation}\lim_{x\to a}\,\sum_{i=1}^N\,f_i(x)=\sum_{i=1}^N\,L_i.\end{equation}

Assume that the follow statements are true:
\begin{eqnarray}
\lim_{x\to a}\,f_1(x)&=&L_1\\
\lim_{x\to a}\,f_2(x)&=&L_2\\
\lim_{x\to a}\,f_3(x)&=&L_3\\
\vdots\\
\lim_{x\to a}\,f_i(x)&=&L_i.\\
\vdots\\
\lim_{x\to a}\,f_N(x)&=&L_N.\\
\end{eqnarray}

Then
\begin{equation}
\lim_{x\to a}\, [ f_1(x)+\cdots +f_i(x)+\cdots +f_N(x) ] = L_1+\cdots +L_i + \cdots +L_N
\end{equation}
which we can write with our sum ($\sum$) symbol as
\begin{equation}
\lim_{x\to a}\,\sum_{i=1}^N\,f_i(x) = \sum_{i=1}^N\,L_i.
\end{equation}