Closed embedded submanifolds of complete Riemannian manifolds are complete

By [F1], $i$ is an immersion. Hence [F2] and [F3] show that $h=i^{\ast }g$ is indeed a Riemannian metric on $S$. By [F7], the connected component $C$ is open in $S$, so it is a connected submanifold and the restriction of $h$ to $C$ is again Riemannian.

Let $M_0$ be the connected component of $M$ containing $C$. If $\gamma$ is a piecewise-$C^1$ curve in $C$, then [F2] gives pointwise equality of speeds and therefore $L_h(\gamma )=L_g(i\circ \gamma )\mathrm{.}$ Every such curve is also an ambient curve in $M_0$. Taking the two infima in [F4] consequently gives $d_g(x,y)\le d_h(x,y)(x,y\in C)\mathrm{.}$

The inequality in step 2.1 makes $(x_n)$ a $d_g$-Cauchy sequence in $M_0$. By the assumed completeness of $M_0$ and [F6], there is $p\in M_0$ such that $d_g(x_n,p)\to 0$.

By [F5], $x_n\to p$ in the manifold topology of $M_0$. If $p\mathrm{\notin }S$, then $M_0\cap (M\setminus S)$ would be an open neighbourhood of $p$ in $M_0$ containing none of the $x_n$, contradicting this convergence. Thus $p\in S$. If $O$ is any neighbourhood of $p$ in $S$, [F1] gives an ambient-open $V$ such that $p\in S\cap V\subseteq O$. Since $V\cap M_0$ is a neighbourhood of $p$ in $M_0$, eventually $x_n\in V\cap S\subseteq O$. Hence $x_n\to p$ in $S$.

The component $C$ is closed in $S$ by [F7]. Since every $x_n$ lies in $C$ and step 4.1 gives $x_n\to p$ in $S$, closedness forces $p\in C$. Because $C$ is also open in $S$, the same convergence is convergence in the manifold topology of $C$.

Apply [F5] to the connected Riemannian manifold $C$. Step 5.1 then gives $d_h(x_n,p)\to 0$. The arbitrary $d_h$-Cauchy sequence $(x_n)$ therefore converges to a point of $C$, so [F6] proves that $(C,d_h)$ is complete. Since $C$ was arbitrary, the componentwise statement and its connected special case follow.