Use induction to prove it...
If would be ?
It's the same for
(we will also require the other root later so we may as well anounce it here)
Now suppose , then let , and we have:
But , so:
and as , we have , and so .
Hence as , for all .
This is greater than for between the roots of that is between .
That is is decreasing while and and as , is a decreasing sequence and also bounded below, so converges.
Now if converges the limit satisfies:
which has roots and But it cannot converge on as the sequence is decreasing so the limit is .