Let $\mathcal{M}$ be a (not necessarily complete) Riemannian manifold. As a result, we cannot claim that closed and bounded subsets are compact. Pick $x \in \mathcal{M}$. Let $r = \text{inj}_{\mathcal{M}}(x)$ be the injectivity radius at $x$. Then $\exp_x$ is continuous and invertible on $B_{T_x\mathcal{M}}(0,r):=\{v \in T_x\mathcal{M}:\|v\|_x < r\}$. Take the geodesic ball $B:=B_{\mathcal{M}}(x,r/2)=\{y \in \mathcal{M}:d(x,y) < r/2\}$. Set $D:=\exp_x^{-1}(B)$. We know that $\exp_x^{-1}(\bar{B})=\bar{D}$. Clearly $\bar{D}$ is compact. Since $\exp_x$ is continuous, we must have $B$ is compact.
Are there any issues with my proof?