I'm aware that the answers to this question already exist on this site I would just like to know if my proof is rigorous enough (or incorrect).
Prove that $\lim_{x \to a}f(x)=\lim_{h \to 0}f(a+h)$. (This is mainlyexercise in understanding what the terms mean)
Proof: The number $\lim_{x \to a}f(x)$ is the number that $f$ approaches at $a,x$ becomes closer and closer to $a$ but it doesn't necessarily equal $a$. As long as $x$ does not equal $a$ it is some distance away from $a$. Let's call this distance $h$. As $x$ approaches $a$, $h$ will get smaller and smaller hence..$$\lim_{x \to a}f(x)=\lim_{h \to 0}f(a+h) $$
