Show that there is no neighborhood of $0$ such that the function $f(x)=x \left[1+\sin^2 \frac{1}{x}\right]$ is increasing
I like what you did. The argument could be more straightforward though.
The sentence
Notice that $\lim_{x \to 0^+} f'(x)$ and $\lim_{x \to 0^-} f'(x)$ don't exist
doesn't support the result to be proved as those limits may not exist while $f^\prime$ may always be positive in a neighborhood of zero.
The only important thing is to get a sequence $\{c_n\}$ converging to zero such that $f^\prime(c_n) \lt 0$ for all $n \in \mathbb N$. And this is what you have for the sequence $\{b_n\}$ you mention.