Difference Between Adjacent Square Roots Converges

Theorem
Let $$\left \langle {x_n} \right \rangle$$ be the sequence in $\R$ defined as $$x_n = \sqrt {n + 1} - \sqrt n$$.

Then $$\left \langle {x_n} \right \rangle$$ converges to a zero limit.

Proof
We have:

$$ $$ $$ $$ $$

But from Power of Reciprocal, $$\frac 1 {\sqrt n} \to 0$$ as $$n \to \infty$$.

The result follows by the Squeeze Theorem.