Prove that if $\displaystyle f(x)<g(x)<h(x)$ and $\displaystyle \lim_{x \to \infty}f(x) = \lim_{x \to \infty}h(x)=L$, then $\displaystyle \lim_{x \to \infty}g(x)=L$.

Can we prove this by contradiction, assuming to the contrary that $\displaystyle \lim_{x \to \infty}g(x)\not = L$?