= Solution
For $p,q\in M$, define the <Riemannian distance> by
$$
\boxed{d_g(p,q)=\inf\{L_g(c):c\text{ is piecewise smooth from }p\text{ to }q\},\qquad
L_g(c)=\int|\dot c(t)|_g\,dt.}
$$
A <connected> <smooth manifold> is locally path <connected>, so any two points are joined by a finite concatenation of coordinate paths. Thus the infimum is finite. It is nonnegative, $d_g(p,p)=0$, and reversing a <curve> proves symmetry. Concatenating two <curves> whose lengths approach their respective infima proves the <triangle inequality>.
For positivity and the topology, choose a coordinate ball about $p$ whose closure is contained in a chart. On its closure there are constants $0<c<C$ with
$$
c|v|_{\mathrm{Eucl}}\le |v|_g\le C|v|_{\mathrm{Eucl}}.
$$
A <curve> staying in the ball has length at least $c$ times its coordinate displacement. A <curve> leaving it must first reach its boundary, costing at least $c$ times the positive coordinate distance from $p$ to that boundary. Therefore a distinct $q$ has $d_g(p,q)>0$, whether it lies inside or outside the ball. For $q$ sufficiently close to $p$, the coordinate line segment gives $d_g(p,q)\le C|x(q)-x(p)|$, while the preceding lower bounds prevent a short <curve> from taking a shortcut outside the chart. These inequalities prove that the <Riemannian distance induces the manifold topology>. In particular it defines a genuine <metric space>.
A <Riemannian manifold> is <geodesically complete> if every maximal affinely parametrized <geodesic> exists for all real parameter values. We prove both implications of the <Hopf-Rinow theorem> rather than assuming global distance minimizers.
First assume completeness as a <metric space>. Let a constant-speed <geodesic> have a finite maximal right endpoint $b$. The bound $d_g(\gamma(t),\gamma(s))\le c|t-s|$ makes $\gamma(t)$ a <Cauchy sequence> as $t\uparrow b$, hence gives a limit $p\in M$. The distance topology just established puts the final portion inside a relatively <compact> coordinate ball about $p$. The positive lower bound on the <Riemannian metric> bounds all coordinate velocity components. The <Christoffel symbols> are bounded on the closure of that ball, and the <geodesic equation>
$$
\ddot x^i=-\Gamma^i_{jk}(x)\dot x^j\dot x^k
$$
then bounds the coordinate acceleration. The velocity has a limit $v$ as $t\uparrow b$. Local existence and uniqueness for the <geodesic equation> with data $(p,v)$ continue $\gamma$ through $b$, contradicting maximality. The same argument applies at a finite left endpoint. Hence \b[metric completeness implies <geodesic completeness>].
Conversely assume <geodesic completeness>. Fix $p,q$, put $r=d_g(p,q)>0$, and choose $0<\epsilon<r$ so small that a closed radius-$2\epsilon$ normal ball about $p$ lies inside a <convex normal neighborhood>. By the <Gauss lemma>, its radial <geodesics> minimize; a <curve> leaving the larger ball cannot improve on a radius-$\epsilon$ radial segment. Thus the radius-$\epsilon$ <geodesic sphere> $S_\epsilon(p)$ is <compact> and consists of points at distance exactly $\epsilon$ from $p$.
The <distance splitting through a small geodesic sphere> has an elementary proof here. Every path from $p$ to $q$ must cross $S_\epsilon(p)$, and its initial portion has length at least $\epsilon$. Hence
$$
r\ge\epsilon+\min_{x\in S_\epsilon(p)}d_g(x,q).
$$
The <triangle inequality> proves the reverse bound. Continuity of $d_g(\cdot,q)$ and <compactness> give a point $x$ achieving the minimum, so $d_g(x,q)=r-\epsilon$. Let $\gamma(t)=\exp_p(tv)$ be the complete unit-speed radial <geodesic> through $x$ at time $\epsilon$.
Consider the closed subset
$$
A=\{t\in[\epsilon,r]:d_g(p,\gamma(t))=t,\quad d_g(\gamma(t),q)=r-t\}.
$$
It is nonempty since $\epsilon\in A$, and has a largest member $t_*$. Suppose $t_*<r$. At $x_*=\gamma(t_*)$, choose a small normal <sphere> of radius $\eta<r-t_*$ and repeat the splitting argument. It supplies a radial endpoint $y$ with
$$
d_g(x_*,y)=\eta,\qquad d_g(y,q)=r-t_*-\eta.
$$
The <triangle inequality> gives $d_g(p,y)\ge t_*+\eta$, while the broken path following $\gamma$ to $x_*$ and then the short radial segment to $y$ has exactly that length. It is therefore minimizing. Its two velocities at the joining point must agree: if unit incoming and outgoing velocities are $u,w$, moving the join in direction $w-u$ changes the sum of lengths by $-|u-w|^2$ to first order, using the <first variation of geodesic energy> or the equivalent unit-speed length formula. A corner would strictly shorten it. Uniqueness of the <geodesic equation> consequently gives $y=\gamma(t_*+\eta)$, contradicting the maximality of $t_*$. Thus $t_*=r$, $\gamma(r)=q$, and $\gamma|_{[0,r]}$ is a <minimizing geodesic>.
This <radial continuation proof of Hopf-Rinow> also proves <compactness> of closed balls. For every $R\ge0$,
$$
\overline B_g(p,R)=\exp_p\{v\in T_pM:|v|\le R\}.
$$
The inclusion from right to left follows from the length of a radial <geodesic>; the reverse inclusion uses the minimizing <geodesic> just constructed. The <Riemannian exponential map> is defined on all of $T_pM$ by <geodesic completeness>, and the closed tangent ball is <compact> in a finite-dimensional <vector space>. Its continuous image is <compact>. Every <Cauchy sequence> is bounded, hence has a convergent subsequence in such a ball, and the Cauchy property makes the whole sequence converge. Therefore \b[<geodesic completeness> implies metric completeness], completing both directions.
For the <hyperbolic plane>, use the <upper half-plane model> with $g=(dx^2+dy^2)/y^2$, $y>0$. Its <geodesic equation> is
$$
x''-\frac{2x'y'}y=0,\qquad y''+\frac{x'^2-y'^2}y=0.
$$
Every nonconstant unit-speed solution is either a vertical <curve> $x=a$, $y=be^{\pm t}$ or, after translating and possibly reversing $t$,
$$
x(t)=a+R\tanh(t-t_0),\qquad y(t)=R\operatorname{sech}(t-t_0),\qquad R>0.
$$
Substitution verifies both equations and $g(\dot\gamma,\dot\gamma)=1$. These <curves> cover every unit initial tangent: a nonvertical tangent at $(x,y)$ determines the circle center $a=x+yy'/x'$ and radius $R=\sqrt{(x-a)^2+y^2}$, and its orientation determines the sign of the parameter. Constant solutions and constant-speed rescalings cover arbitrary initial velocities. In every case $y(t)>0$ for all finite $t$, and the solution is defined for every real $t$. Thus \b[the real <hyperbolic plane> is <geodesically complete>], directly, and the proved <Hopf-Rinow theorem> also makes its distance complete.
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