How to prove that an integral converges

Question

Let $(a_n)$, $(M_n)$ be sequences of positive real numbers such that ${a_n} \downarrow 0$, ${M_n} \uparrow \infty$ as $n\to\infty$. Let $\alpha>0$ and $\beta>1$. How to prove the following statement $$\int\limits_{ - \infty }^\infty {{e^{ - {{\left| {{a_n}u + {M_n}} \right|}^\alpha }}}{{\left| u \right|}^\beta }du} < \infty$$ for all $n$ large enough. Thank so much for heplping.

Answer 1

$$\frac{|u|^\beta}{e^{|a u + M|^\alpha}} < \frac{1}{|u|^2}$$ for large $|u|$.