Proof that the sequence $s_n = \frac{1}{n}$ converges to $0$ 
Prove that the sequence $s_n = \frac{1}{n}$ converges to $0$.

I am writing this proof in order to help other people to understand better how to prove if a sequence converges and in particular why $s_n = \frac{1}{n}$ converges to $0$. 
Please, make me know if this proof is correct and how can I improve it.

Another way to handle it: substitute the sequence at hand and the expected limit in the definition.

The sequence $\dfrac1n$ converges to $0$, if for any $\epsilon >0$, there exits a number $N$, such that $n > N$ implies $\left|\dfrac1n - 0\right| < \epsilon$.

Indeed the inequality holds for all $n>\dfrac1\epsilon$ so that $N=\left\lfloor\dfrac1\epsilon\right\rfloor$ satisfies the definition.

The sequence $\dfrac1n$ converges to $0$, because for any $\epsilon >0$, $n > N=\left\lfloor\dfrac1\epsilon\right\rfloor$ implies $\left|\dfrac1n - 0\right| < \epsilon$.

