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:

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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.