Let $\left\{ u _ { n } \right\} _ { n \geq 1 }$ be a sequence of real numbers defined as $u _ { 1 } = 1$ and
$$u _ { n + 1 } = u _ { n } + \frac { 1 } { u _ { n } } \text { for all } n \geq 1 .$$
Prove that $u _ { n } \leq \frac { 3 \sqrt { n } } { 2 }$ for all $n$.