Let $(M,g)$ be a Riemannian manifold, $K\subset M$ a compact subset, and $$\widehat{K}=\{v_q\in TM:q\in K,v_q\in T_qM,|v_q|_g\leq 1\}$$ How to show that $\widehat{K}$ is compact?
I have tried to modify the proof that the product of two compact spaces is compact. In particular, I am trying to prove the tube lemma for tangent bundles. However, I failed. Did I miss something obvious?
Any help?
Your strategy looks reasonable. Cover $K$ by trivializing neighborhoods of $TM$, arranging that the closure of each neighborhood is contained in some trivializing neighborhood, and use compactness to select a finite subcover $(U_{i})_{i=1}^{N}$. The restriction of $TM$ to $\overline{U}_{i}$ is trivial, so the unit ball subbundle is compact (because $\overline{U}_{i}$ and the unit ball are compact). Therefore $\widehat{K}$ is contained in a finite union of compact sets.
