Could someone give me guidance for the proof of the following problem?
For $n=1,2,3,…$ and $x$ is a real number, put
$f_n(x)=\frac{x}{1+n x^2}$
Show that ${f_n(x)}$ converges uniformly to a function $f$.
It is a problem from Baby Rudin chapter 7.
The proof for this problem, which is provided from Roger Cookes solution manual (https://minds.wisconsin.edu/bitstream/handle/1793/67009/rudin%20ch%207.pdf?sequence=5&isAllowed=y) says,
From Schwartz inequality,
$\mid f_n(x)\mid \leq \frac{\mid x\mid}{\sqrt{2}n\mid x\mid} \mid$
How does he do this manipulation?
I can only get
$\mid f_n(x)\mid^2\leq \frac{\mid x\mid^2}{\mid 1+n x\mid^2}$.
From the Schwartz inequality.
Any comments would be very appreciated!
Best Answer
You can use $AM-GM$, this means $$1+nx^2\geq2\sqrt{n}|x|$$