Are the following statements equivalent?
Statement 1
For any $\epsilon>0$, there exists $\delta>0$ such that for all real number $x$, if $|x-x_0|<\delta$, then $|f(x)-f(x_0)|<\epsilon$
Statement 2
There exists $\delta>0$ such that for any $\epsilon>0$, for all real number $x$, if $|x-x_0|<\delta$, then $|f(x)-f(x₀)|<\epsilon$
If so, can the second statement also be used as a definition for limits?