$C^\infty$ function contained in compact set has convergent sequence on it's image

Question

Let $f \colon [a, b) \rightarrow X$ be a $C^\infty$ function and $X$ a topological space. Suppose the image of $f$ is contained in a compact set. Prove that there is a sequence $\{x_n\} $ in $[a, b)$ that converges to $b$ and that $\{\alpha(x_n)\}$ converges.

Any hints? Thanks!

Answer 1

Choose any sequence $(x_n)$ in $[a,b)$ converging to $b$. Then the sequence $(f(x_n))$ is contained in the image of $f$, a compact set. Use compactness to find a subsequence of $(x_n)$ with the desired properties.