I am looking for a function $f: R\rightarrow R$ and a sequentially compact $K\subset R$ such that the inverse $f^{-1}(K)$ is not sequentially compact.
I decided to choose $f(x) = sin(x)$, but I'm not sure what $K$ could be such that $f^{-1}(K)$ is not sequentially compact.
Are there any ideas about what $K$ can be? I'm having a hard time grasping the concept of sequentially compact.
I'm thinking $K=[-\pi/2,\pi/2]$
You can get even simpler than this, when in doubt look at the extreme cases. Define $f(x)=0$ to be the constant function. Then {0} is sequentially compact, but $f^{-1} (\{ 0\})=\mathbb R$, which isnt.