Is the sequence of functions $f_n(x)=\frac{nx}{1+nx^2}$ a Cauchy sequence in $C([0,1])$?
I'm a little lost as to how to go about this. I thought I could just check $|f_n(x)-f_m(x)|$ and show that it is/isn't $<\epsilon$, for $\epsilon$ small, but I confused myself somewhere. Any help/hints would be greatly appreciated.
No, it is not a Cauchy sequence in the space $C[0,1]$ with the supremum norm. If it were, then as the space is complete, it would have converged uniformly to a continuous function, and in particular, it would have converged point-wise to a continuous function, but clearly for $x\neq 0$,
$$\lim_{n\to\infty}\frac{nx}{1+nx^2}=\frac{1}{x},$$ and there is no continuous function on $[0,1]$ which is equal to $1/x$ for $x\neq 0$.