Is this line of reasoning legitimate?

Let $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = x^5$. $f$ is not uniformly continuous. 

Suppose it were. Fix $\epsilon>0$ and the associated $\delta$, and choose $x,~ y>0, ~x>y$, st $|x-y|< \delta$.

Then, $|x-y|<\delta \Rightarrow |x^5 - y^5|<\epsilon ~\forall x, y$

Fix $|x-y| = C$. Then,

$$|x^5 - y^5| = |x-y||x^4 + x^3y + x^2y^2 + xy^3 + y^4|= $$

$$C|x^4 + x^3y + x^2y^2 + xy^3 + y^4|<\epsilon$$.

This is valid for all $x, y$ sufficiently close, but because the left hand side increases monotonically, it can be made arbitrarily large while keeping $|x-y|<\delta$, so this is a contradiction. Therefore f is not uniformly continuous.

I'm pretty sure this is logically sound *kinda* and I think my intuition (that because f is increasing increasingly fast without bound, no set of fixed-sized intervals in the domain map to a set of bounded intervals in the image) but I think I can get away with a lot less. Any advice?