Pinching Theorem of Function Limits

Suppose that
$$\begin{equation} \lim_{x\to a}\,u(x)=L\quad\mbox{and}\quad\lim_{x\to a}\,g(x)=L. \end{equation}$$
Furthermore, assume that $u(x)\leq f(x)\leq g(x)$ for some values of $x$ such that $0\lt |x-a|\lt p$ where $p\gt 0$. In other words, $u(x)\leq f(x)\leq g(x)$ is guaranteed to be true around some neighborhood of $a$. Prove that
$$\begin{equation} \lim_{x\to a}\,f(x)=L. \end{equation}$$