Showing a sequence tends to infinity

Define a sequence recursively by .

Given , show lim .

Note: It can easily be shown that so just assume this is given.

Let .
We have the following reccurence for sequence :

But for this sequence explicit formula can be easily obtained:

Because the sequence is not convergent.

Hey, sorry for the late response, thanks though.

So I can just assume and just say since,

for all ?