Let us consider a sequence $x_n$. Now let it converge to a limit $L$. Now which one of the following is the correct definition of convergence?
A sequence $x_n$ is said to be convergent to a limit $L$ if given any integer $n$ there exists a positive real number $\epsilon$ such that for all $M\gt n$, $|x_M-L|\lt\epsilon$.
A sequence $x_n$ is said to be convergent to a limit $L$ if given any real positive number $\epsilon$ there exists an integer $n$ such that for all $M\gt n$, $|x_M-L|\lt\epsilon$.
If the two definitions are equivalent then how to prove it?