Prove that: ⌊√(n)+√(n+1)⌋=⌊√(4n+2)⌋, for all positive integer n.

Prove that: $\left\lfloor{\sqrt{n}+\sqrt{n+1}}\right\rfloor= \left\lfloor{\sqrt{4n+2}}\right\rfloor$, for all $n\in N$.

My solution:

Step 1:

Note that:

$4n^2+4n\lt 4n^2+4n+1\lt 4n^2+8n+4$