How to formally prove that if $\lim \limits_{n \to \infty}a_n=\infty$, then $\lim \limits_{n \to \infty}\frac{1}{a_n}=0$.

How to formally prove that if $\lim \limits_{n \to \infty}a_n=\infty$, then $\lim \limits_{n \to \infty}\frac{1}{a_n}=0$.

I got confused for how to mix between convergence to infinity and having finite limit. how do I write a proof to link between two things? ( by using $\epsilon$) and not just giving an example.

my proof:

let $M= \frac{1}{\epsilon}$. there exists such a $N\in\mathbb N$ s.t for every $n>N$, it is true that: $$a_n>\frac{1}{\epsilon}$$

we should prove that for every $\epsilon>0$ there exists such a $N\in\mathbb N$, s.t for every $n>N$ $$\left|\frac1{a_n}\right|<\epsilon$$

so we take the N that satisfies the first conclusion, and that will mean for every $n>N$$\left|\frac1{a_n}\right|<\epsilon$$ 1 Answer Use the strict definition. That is, use the fact that: For every$M\in \mathbb R$, there exists such a$N\in\mathbb N$that for every$n>N$, it is true that $$a_n>M$$ And prove the fact that: For every$\epsilon\in \mathbb R$, there exists such a$N\in\mathbb N$that for every$n>N$, it is true that $$\left|\frac{1}{a_n}\right|<\epsilon$$ To do this: 1. Take an arbitrary$\epsilon$. 2. Try to see how large the value of$a_n$must be in order for the value of$\left|\frac{1}{a_n}\right|$to be below$\epsilon$. Call that value$C$. 3. You can now find the value of$N$for which$a_n>C$for all$n>N$, meaning that$\left|\frac1{a_n}\right|<\epsilon$for all$n>N$. • can I take$M= \frac{1}{\epsilon}$according to the first definition?. and because after certain$n$,$a_n >0$, and I can take it out of the absolute value and find the relation between$M$and$\epsilon$. – Firas Ali Abdel Ghani Feb 16 '15 at 10:22 • @FirasAliAbdelGhani I think you have the correct idea, though the way you phrased it is a little awkward. If you write your proof into the question by editing it, I can tell you exactly if it is ok. – 5xum Feb 16 '15 at 10:27 • I edited. what do you think ? – Firas Ali Abdel Ghani Feb 16 '15 at 10:54 • @FirasAliAbdelGhani Looks fine to me. Just add 2 things: First of all, before saying "let$M=1/\epsilon$", you need to select the value of$\epsilon$. It is better to start your proof with "let$\epsilon > 0$" and go from there. And second, while it is simple, you have not pointed out the link between the fact that$a_n>1\epsilon$and the fact that$\left|\frac1{a_n}\right|<\epsilon\$. I think you know these things, but without writing them down, your professor may not know that you do. – 5xum Feb 16 '15 at 11:17
• I think that you're totally right about that. thanks for helping!! – Firas Ali Abdel Ghani Feb 16 '15 at 11:40