Formal proof of $\lim_{x\to a}f(x) = \lim_{h\to 0} f(a+h)$ $$ \lim_{x\to a} f(x) = \lim_{h\to 0} f(a+h) $$


How do I write a formal proof of it?
Suppose $\displaystyle \lim_{x \to a}f(x) = c$ and let $\epsilon &gt; 0$, then there is $\delta &gt;0$ such that $$|x-a|&lt;\delta \implies |f(x)-c | &lt; \epsilon$$
now note that $$|h| = |(a+h)-a|$$ and so
$$|h|&lt; \delta \implies  |f(a+h)-c | &lt; \epsilon$$
which shows that $\displaystyle \lim_{h \to 0}f(a+h) = c$. It follows that
$$\displaystyle \lim_{x \to a}f(x) = c= \displaystyle \lim_{h \to 0}f(a+h).$$
Pose $h=x-a$, then $x=a+h$ and when $x\to a$, there is $h\to 0$. Then, 

$$\lim_{x\to a}f(x)=\lim_{x-a\to 0}f(x-a+a)=\lim_{h\to 0}f(a+h)$$