# Convergent Sequences problem

• pbxed

## Homework Statement

Let f: N -> N be a bijective map. for n Є N

a sub n = 1 / f(n)

Show that the sequence (a sub n) converges to zero.

## The Attempt at a Solution

Basically I have been stuck on this problem for hours now and have read through my notes and the course textbook numerous times and am still not getting anywhere. I sort of get how to apply the definition of a convergent sequences to a few other questions but my understanding of this topic is clearly pretty poor.

Can someone give me a hint either as a question to get me started or some material available that would clarify this for me ?

So you'll want to show that

$$\forall \epsilon >0~\exists n_0~\forall n>n_0:~|1/f(n)|<\epsilon$$

This is equivalent with (take $$N=1/\epsilon$$):

$$\forall N>0~\exists n_0~\forall n>n_0:~|f(n)|>N$$

So you'll have to show that, from a certain moment on, the sequence gets bigger then N (and this forall N). But since f is a bijection, it is true that, from a certain $$n_0$$, f has already assumed all elements smaller then N. So from that moment on, f is always larger then N. Which we needed to prove.

Ï hope you understand my ramblings is it rude of me to say that I don't really understand your reply and ask for further clarification, either from yourself on someone else?