We prove the key implications.

(3) $\Rightarrow$ (4) and the existence of minimizing geodesics. Fix $p \in M$ and assume $\exp_p$ is defined on all of $T_pM$. Let $q \in M$ with $d(p,q) = r$. We show there exists a minimizing geodesic from $p$ to $q$.

Choose a small $\epsilon > 0$ such that $\exp_p$ is a diffeomorphism on $B_\epsilon(0) \subset T_pM$. On the geodesic sphere $S_\epsilon = \exp_p(\{v : |v| = \epsilon\})$, by compactness, there exists a point $x_0 \in S_\epsilon$ minimizing $d(x_0, q)$.

Claim: $d(p, q) = \epsilon + d(x_0, q)$.

Any path from $p$ to $q$ must cross $S_\epsilon$, and the shortest way to reach $S_\epsilon$ from $p$ is the radial geodesic of length $\epsilon$. So $d(p,q) \geq \epsilon + \min_{x \in S_\epsilon} d(x, q) = \epsilon + d(x_0, q)$. The reverse inequality follows from the triangle inequality.

Let $\gamma$ be the unit-speed geodesic from $p$ through $x_0$. Define $A = \{t \in [0, r] : d(p, \gamma(t)) + d(\gamma(t), q) = r\}$. Then $\epsilon \in A$ and $A$ is closed. The key step: if $t_0 \in A$, repeat the argument at $\gamma(t_0)$ to show $t_0 + \delta \in A$ for small $\delta > 0$. This shows $A = [0, r]$, so $\gamma(r)$ satisfies $d(p, \gamma(r)) = r$ and $d(\gamma(r), q) = 0$, hence $\gamma(r) = q$, and $\gamma$ is minimizing.

For the Heine-Borel property: the closed ball $\bar{B}_r(p) = \{q : d(p,q) \leq r\} = \exp_p(\{v : |v| \leq r\})$ is the continuous image of a compact set, hence compact. Any closed bounded set is a closed subset of some $\bar{B}_r(p)$, hence compact.

(4) $\Rightarrow$ (1): Heine-Borel implies completeness in any metric space.

(1) $\Rightarrow$ (2): If a geodesic $\gamma$ cannot be extended past time $T$, then $\gamma(t_n)$ for $t_n \to T$ is Cauchy (since $d(\gamma(s), \gamma(t)) \leq |s - t|$), so it converges to a limit $p_0$. By the local existence of geodesics near $p_0$, we can extend $\gamma$ past $T$, contradiction.

(2) $\Rightarrow$ (3): Immediate from the definition of the exponential map. $\blacksquare$