Proving $\lim_{n\to\infty} 2/n = 0$ using $\epsilon-N$ verification.

Question

I need help checking my proof showing that $\lim_{n\to\infty} 2/n = 0$.

Proof:

Let $\epsilon > 0$, and choose $N = \lceil{2/\epsilon}\rceil + 1$. Then for all $n > N$, we have $\left|\frac{2}{n} - 0\right| = 2/n$. But since $n > N = \lceil2/\epsilon\rceil + 1 > \lceil2/\epsilon\rceil \geq 2/\epsilon$, we have $\epsilon > 2/n$, from which it follows that $2/n < \epsilon$, as we desired to show.

Answer 1

Take an $\epsilon>0$.

We look for $N\in \Bbb N$ such that

$$n>N \implies \frac 2n<\epsilon$$

or

$$n>N \implies n>\frac 2\epsilon$$

so we can take $$N=\lfloor \frac 2\epsilon \rfloor +1.$$