Uniform convergence of a function defined as an integral

Question

I have to prove that this function : $$f_n(x) = \int\limits_0^n e^{-\left(1+x^2\right)t^2}\,\mathrm{d}t$$ converges uniformly.

In the solution of this exercise, they use a sequence $a_n = \int_{n-1}^n e^{-t^2}dt $ and they prove that $|f_n(x)-f_{n-1}(x)|<a_n $. Then they use that to conclude on the convergence of $f_n$.

I don't really understand the reasoning and why it works. Could you please help me?

Answer 1

$\sum a_n=\int_0^{\infty} e^{-t^{2}} dt <\infty$. Apply M-test to conclude that $\sum [f_n(x)-f_{n-1}(x)]$ converges uniformly. But the $n-$th partial sum of this series is nothing but $f_n(x)$ so this proves that $(f_n(x))$ converges unifromly. [Here I define $f_0$ as $0$].