I've been given an outline to prove the maximum principle for the heat equation with the growth condition $|u(x,t)| \le Ae^{\alpha|x|^2}$. The first step is to show:
If $T$ is small such that $$4\alpha T<1$$ Then there exists $\epsilon>0$ such that $$4\alpha (T+\epsilon)<1$$
I don't know where to begin because this sounds impossible. Wouldn't you get
$$4\alpha (T+\epsilon)<1+4\alpha \epsilon$$
Any help is appreciated
Note that $A := 1 - 4\alpha T > 0$, then take $\epsilon > 0$ small enough such that $A - 4\alpha\epsilon > 0$ and you'll have your result. (Edit: Take, for instance, any $0 < \epsilon < \frac{A}{8\alpha}$).