real analysis – Show that \$ f_ {n} (x) = frac {x} {1 + nx ^ {2}} \$ converges uniformly.

CA watch $$f_ {n} (x) = frac {x} {1 + nx ^ {2}}$$ converges uniformly.

I looked at Rudin's answer to this proof:

I do not understand the part that I have squared in red. How to use Cauchy-Schwarz inequality to get this result? I am only used to using inequality when it comes to convergent series.