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, $\dfrac 1 {\sqrt n} \to 0$ as $n \to \infty$.

The result follows by the Squeeze Theorem.