finding a minorant to $(\sqrt{k+1} - \sqrt{k})$ Need help finding a minorant to
$(\sqrt{k+1} - \sqrt{k})$ which allows me to show that the series $\sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})$ is divergent.

If you do want a minorant, then you can use
$$
\sqrt{k+1}-\sqrt k = \int_k^{k+1} \frac1{2\sqrt x} dx > \int_k^{k+1} \frac1{2\sqrt {k+1}} dx = \frac1{2\sqrt {k+1}}.
$$
