Show that $\lim_{n\rightarrow\infty} \frac{1}{x_{n}}=0$ [proof verification]

Question

I am asked to prove that if a sequence $(x_n)$ increases without a bound, then $$\lim_{n\rightarrow \infty}\frac{1}{x_n}=0.$$ Since $(x_n)$ inceases without a bound, then for every $M\in\mathbb{R}_+$ there exists $n_{\epsilon}\in\mathbb{N}$ such that $x_n>M$ when $n>n_{\epsilon}$. Let $\epsilon=\frac{1}{M}$. Now $|\frac{1}{x_n}-0|=\frac{1}{x_n}<\frac{1}{M}=\epsilon, $when $n>n_{\epsilon}$. Thus $\lim_{n\rightarrow \infty}\frac{1}{x_n}=0.$ Is this correct and complete?

Answer 1

It's almost correct. You can't say “let $\varepsilon=\frac1M$”. In fact, for every $\varepsilon>0$, what you do is to take $M=\frac1\varepsilon$. After this, what you did (that is, to use what is known about the sequence $(x_n)_{n\in\mathbb N}$) is correct.