Upper bound on $\chi(G)$ for a triangle-free graph

Here's a sketch:

Start with a lemma: Every triangle-free graph contains an independent set of size $\sqrt{n}$. This can be proven by degree considerations: If there exists a vertex of degree $\geq \sqrt{n}$, consider it's neighbors. Otherwise the maximum degree is at most $\sqrt{n}-1$ and apply the greedy algorithm.

To prove the result, apply induction. Let $S$ be an independent set of size $\sqrt{n}$, and let $H=G\setminus S$. Then $H$ is triangle free and has at most $n-\sqrt{n}$ vertices, and so can be properly colored using at most $2\sqrt{n-\sqrt{n}}+1$ colors. Conclude that $G$ can be properly colored using $2\sqrt{n}+1$ colors.