My question is about my proof: whether a specific step I'm making is valid. My proof is as follows:
Suppose that $\lim_{x\to a} f(x)$ exists, and that it is equal to $L$. Then, for every $\epsilon > 0$ there exists a $\delta > 0$ such that $|x - a| < \delta$ implies $|f(x) - L| < \epsilon$.
So write $h = x - a$, and fix $\epsilon' > 0$. Then, there exists a $\delta' > 0$ such that $|x - a| = |h| < \delta'$ implies $|f(x) - L| = |f(a + h) - L| < \epsilon'$, so $\lim_{h\to 0} f(a+h) = L$.
My proof for the other direction is very similar, so I won't post it, but my question is about the step "So write $h = x - a$". Is this a valid step? The reason I'm asking is that I'm not sure that I can just define $h$ however I like.