# Thread: Proving limits help please???

1. ## Proving limits help please???

Hi, can anyone give me a hand with this question, i honestly dont know what to do!!

Prove that lim (n tends to infinity) 1/an = 0 when an is a sequence with an > 0 for n being natural, and lim (n tends to infinity) an = infinity.

2. Originally Posted by Bexii
Hi, can anyone give me a hand with this question, i honestly dont know what to do!!

Prove that lim (n tends to infinity) 1/an = 0 when an is a sequence with an > 0 for n being natural, and lim (n tends to infinity) an = infinity.
1 divided by a very big number is a very small number.. so if you divide something by infinity, you will get infinite small number = 0.

3. i was told i had to use definitions in my answer...

4. $\displaystyle \forall \,\epsilon >0,\,\exists \,k\in \mathbb{N}\mid n\ge k\implies \left| \frac1n\right|<\epsilon .$

But this is quite straightforward, so we get that $\displaystyle \frac1n<\epsilon$ which means that $\displaystyle n>\frac1\epsilon,$ so for any $\displaystyle k\in\mathbb N$ being greater than $\displaystyle \frac1\epsilon,$ the above condition is fulfilled. For example, it's enough to take $\displaystyle k=\left\lfloor \frac{1}{\epsilon} \right\rfloor +1.$