proof lim x-a f(x)= lim x-0 f(x+a) ( duplicate)

Question

i guess the proof here(Formal proof of $\lim_{x\to a}f(x) = \lim_{h\to 0} f(a+h)$) is something wrong, its too easy and i thought i couldnt change things like the one most voted

Answer 1

By the formal substitution $x\leftrightarrow a+h$, $$\lim_{x\to a}f(x)=L\iff\forall\,\epsilon>0:\exists\,\delta>0:|x-a|<\epsilon\implies|f(x)-L|<\delta$$

is strictly equivalent to

$$\lim_{a+h\to a}f(a+h)=L\iff\forall\,\epsilon>0:\exists\,\delta>0:|a+h-a|<\epsilon\implies|f(a+h)-L|<\delta$$

or after simplification $$\lim_{h\to 0}f(a+h)=L\iff\forall\,\epsilon>0:\exists\,\delta>0:|h|<\epsilon\implies|f(a+h)-L|<\delta.$$