real analysis - $lim _{xrightarrow c}f(x)=L$ if and only if $lim_{xrightarrow0}f(x+c)=L$

Let $f:= \mathbb{R}\to\mathbb{R}$ and let $c\in\mathbb{R}$. Show that $\lim _{x\rightarrow c}f(x)=L$ if and only if $\lim_{x\rightarrow0}f(x+c)=L$.

From the definition of limit, we get that it is enough to show:

$\forall$ $\varepsilon>0$ $\exists$ $\delta>0$ s.t. if $|x-c|<\delta$ then $|f(x)-L|<\varepsilon$
$\Leftrightarrow$ $\forall$ $\varepsilon_0>0$ $\exists$ $\delta_0>0$ s.t. if $|x|<\delta_0$ then $|f(x+c)-L|<\varepsilon_0$

I can replace $x$ by $x+c$ everywhere in statement for the if $(\Rightarrow)$ part. But, I am not sure this is the correct method. What I need to do is manipulate the inequalities in each to get the other. But, I am not sure how to proceed with that.

Answer

Given $\varepsilon>0$, use the $\delta$ from the definition of $\lim_{x\to c}f(x)=L$. If $|x|<\delta$, then $|(x+c)-c|<\delta$, so $|f(x+c)-L|<\varepsilon$. Thus $\lim_{x\to0}f(x+c)=L$.