$\lim \limits_{n \to \infty}( \cos{\sqrt{n+1}} - \cos{\sqrt{n}})$ Please, help me with this sequence... I am working really hard on sequences, and I struggle... Please, explain. You will have my up vote and eternal gratitude. 

Another way of solving would be to "convert" to real numbers and use the Mean Value Theorem. Let $f(x) = \cos(\sqrt{x})$, which is obviously differentiable. Then we have by MVT that $\exists c \in (x,x+1)$ s.t.
$$\frac{\cos(\sqrt{x+1}) - \cos(\sqrt{x})}{x+1-x} = f'(c) = - \frac{\sin(\sqrt{c})}{2\sqrt{c}}$$
Now taking limit $x \to \infty$ from both sides (this gives us $c \to \infty$):
$$\lim_{x \to \infty} \cos(\sqrt{x+1}) - \cos(\sqrt{x}) = \lim_{c \to \infty} - \frac{\sin(\sqrt{c})}{2\sqrt{c}} = 0$$
Now as the limit exists for real numbers it must hold for natural numbers too.
