Prove uniform convergence of the series

[Like Tony Stark]

[MATH]\sum_{n=0}^\infty \frac{x^n}{n!} [/MATH]
I proved it using the M-test: the series is always smaller or equal to [MATH]\frac{2^n}{n!}[/MATH]. This series can be proved to be convergent by D'alembert test, so the other one must converge uniformly.

But I want to understand how the definition works, so how could I prove it to be uniformly convergent by definition?

Thanks