You left out a term in the last line. It should read
What you need to show now is that:
or equivalently, that
Squaring both sides (both sides are positive, as ) we get
I'll let you conclude
Therefore, the following should be true
We try to prove this using the first statement, since that will link every pair of adjacent natural numbers in a chain.
The adjacent-term chain is established.
Finally prove the inequality is true for an initial value of n.
Then it's true for that n and all following n.