Proving limits using u-substitution

Question

I'm trying to prove that u-substitution is allowed in limits.

How might one prove that if $w = \displaystyle\lim_{n \to a} g\left(n\right)$ then:

$ \displaystyle\lim_{n \to a} f\left(g\left(n\right)\right) = \lim_{u \to w} f\left(u\right) $