"\(\displaystyle \lim_{x\to a} f(x)= L\)" means "Given \(\displaystyle \epsilon> 0\) there exist \(\displaystyle \delta> 0\) such that if \(\displaystyle |x- a|< \delta\) then \(\displaystyle |f(x)- L|< \epsilon\)".
Here, f(x)= x and L= a so that becomes "Given \(\displaystyle \epsilon> 0\) there exist \(\displaystyle \delta> 0\) such that if \(\displaystyle |x- a|< \delta\) then \(\displaystyle |x- a|< \epsilon\)".
Comparing the two inequalities you should see they are the same- you just have to take \(\displaystyle \delta= \epsilon\).
Typically that observation is enough but if you wanted to give a "rigorous" proof you would say
"Given \(\displaystyle \epsilon> 0\) if \(\displaystyle |x- a|< \epsilon\) then \(\displaystyle |f(x)- L|= |x- a|< \epsilon\)".
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