I want to prove that if $0 \le |x| < a$ then:
$$
\sum\limits\_{n=1}^{\infty} \frac{|x|^{n-1}}{n!} < 
\sum\limits\_{n=1}^{\infty} \frac{a^{n-1}}{n!}
$$

I initially attempted to use induction to prove that 
$\forall n > 1, \frac{|x|^{n-1}}{n!} < \frac{a^{n-1}}{n!}$,
but then learned that it is [not the appropriate way of proving this.](https://math.stackexchange.com/questions/98093/why-doesnt-induction-extend-to-infinity-re-fourier-series)

Am I even correct to assume that the above statement can be proven by showing that $\frac{|x|^{n-1}}{n!} < \frac{a^{n-1}}{n!}$ holds $\forall n > 1$? What is the proper way of approaching this?

Does it suffice to show that $\lim\limits\_{n \to \infty} \frac{|x|^{n-1}}{a^{n-1}} < 1$?

Thanks in advance.