Prove that $\lim_{x\rightarrow 0}|x|=0$ Please check my proof 
There exist $\delta $such that $0<|x|<\delta $ imply $|x|< \epsilon $
Then choose $\delta =\epsilon $
for x is real number and $0<|x|<\delta \rightarrow |f(x)-0|=|0-0|< \delta =\epsilon $
Then limit is 0

No, let $\varepsilon>0$ and $\delta=\varepsilon$.
if $|x|<\delta$, then,
$$|f(x)-0|=|x|<\delta=\varepsilon,$$
and the claim follow.
