Please help me check my proof that the transcendental numbers are dense in $\mathbb{R}$

You can remove almost all of the words in your proof! By including so many details, you make the proof hard to read. This is all you need:

Let $a,b \in \mathbb{R}$ s.t. $a<b$. By the density of the rational numbers in the reals, $\exists\, q\neq0 \in \mathbb{Q}$ s.t. $\frac{a}{\pi} < q < \frac{b}{\pi}$ and therefore $a < \pi q < b$. Since $\pi q$ is transcendental, we are done.

If your reader doesn't believe that $\frac{a}{\pi} < q < \frac{b}{\pi}$ implies $a < \pi q < b$, or that $\pi q$ is transcendental when $q\neq0$, then it's their responsibility to educate themselves.