I need to prove this result, and although I can see that it is true, I cannot see how to prove it mathematically..rather than just putting in numbers!
sqrt(n+1) - sqrt(n) tends to 0
It depends on how rigorous a proof you require .... A simple approach is to multiply the expression by $\displaystyle \sqrt{n + 1} + \sqrt{n}$ and simplify. You get $\displaystyle \frac{1}{\sqrt{n + 1} + \sqrt{n}}$ and the limiting value is 'obvious' (you could use the sandwich/pinching/squeeze theorem).