To get at something like this, you have to start with the innermost square root and work your way outwards.
So start by looking at . Since , it follows that .
Now take the next square root: . Since , we can repeat the previous argument and get .
Take the next square root: . (The last bit follows from the facts that and .)
Continuing in this way (using induction if you want to prove it properly), you see that for k=1,2,...,n-2, .
For k=n-2, this says that . Therefore . (You need to be a bit careful at this stage, because when k=n-1 it's no longer true that .)
Finally, . It's then easy enough to check that the right-hand side is less than n.