I have a homework question: Is the sequence $$ f_n(x) = \frac {nx}{1+(nx)^2} $$ Cauchy in the space $ C^0([−1, 1], \mathbb {R} ) $ with the metric induced from the sup norm? Could you please write down all of the relevant steps.
Also I do know what it means for a sequence to be Cauchy but I just haven't tried to do it with the sup norm before and I'm not sure if I'm doing it right.
My attempt:
So firstly I look at the sup norm relative to $f_n$, i.e. $$ d_\infty=\sup_{x} |f_n| $$ To do this I look at the maxima of $f_n$. i.e. when $\frac{d}{dx} (f_n) =0$ $$ 0=\frac{d}{dx} \left(\frac{nx}{1+(nx)^2}\right) = \frac{n-n^3x^2}{(1+(nx)^2)^2} $$ Thus at $ x=\frac{1}{n} $ is a local minimum $ \frac{-1}{2}$ and at $ x=-\frac{1}{n} $ is a local maximum $ \frac{1}{2}$.
But I don't really know how to use this with the definition of a Cauchy sequence?
$f_n$ converges pointwise to $0$. Since $f_n(1/n)=1/2$, the convergence is not uniform. It follows that $\{f_n\}$ is not a Cauchy sequence for the distance $d_\infty$.
You can also prove it directly like this: $$ d_\infty(f_n,f_{2n})\ge\Bigl|\,f_n\Bigl(\frac1n\Bigr)-f_{2n}\Bigl(\frac1n\Bigr)\Bigr|=\Bigr|\frac12-\frac25\Bigr|=\frac{1}{10}. $$