Proof of Every Sequence in a Compact Subset of a Metric Space Has a Cluster Point There

Suppose, for contradiction, that no point of $K$ is a cluster point of $(x_m)_{m\in\mathbb{N}}$. Note first that $x_1\in K$ by hypothesis, so $K$ is nonempty.

Step 1 (a family of open sets that the sequence eventually avoids). Let

$$\mathcal{U}=\{U\subseteq X:\ U\in\mathcal{T}_d\ \text{and there is}\ N\in\mathbb{N}\ \text{with}\ x_m\notin U\ \text{for every}\ m\in\mathbb{N}\ \text{satisfying}\ N\le m\}.$$

We regard $\mathcal{U}$ as a family of subsets of $X$ indexed by $\mathcal{U}$ itself, the member indexed by $U$ being $U$.

Step 2 ($\mathcal{U}$ covers $K$). Let $x\in K$. By assumption $x$ is not a cluster point of $(x_m)$, so by Cluster Point of a Sequence in a Metric Space there are a real number $\varepsilon>0$ and $N\in\mathbb{N}$ such that no $m\in\mathbb{N}$ with $N\le m$ satisfies $d(x_m,x)<\varepsilon$. Consider the open ball $B_d(x,\varepsilon)$, which is open in $(X,d)$ by Open Ball in a Metric Space is Open, hence lies in $\mathcal{T}_d$. If $m\in\mathbb{N}$ satisfies $N\le m$, then $d(x_m,x)<\varepsilon$ fails, and by the symmetry axiom of a metric $d(x,x_m)=d(x_m,x)$, so $d(x,x_m)<\varepsilon$ fails and therefore $x_m\notin B_d(x,\varepsilon)$. Hence $B_d(x,\varepsilon)\in\mathcal{U}$. Moreover $d(x,x)=0<\varepsilon$ by the identity-of-indiscernibles axiom of a metric, so $x\in B_d(x,\varepsilon)$. Since $x\in K$ was arbitrary, $K$ is contained in the union of the members of $\mathcal{U}$, so $\mathcal{U}$ is an open cover of $K$ in $(X,\mathcal{T}_d)$.

Step 3 (a finite subcover). The subset $K$ is compact in $(X,\mathcal{T}_d)$, so statement 1 of Compact Subset Criterion via Open Covers in the Ambient Space holds for $K$, and therefore so does statement 2 of that theorem. Applied to the open cover of Step 2, it yields a finite subset $J\subseteq\mathcal{U}$ with

$$K\subseteq\bigcup_{U\in J}U.$$

Since a union indexed by the empty set is empty and $K$ is nonempty, $J$ is nonempty. Being finite and nonempty, $J$ has $n$ elements for some natural number $n$, so there is a bijection $\beta:[n]\to J$, where $[n]$ is the initial segment determined by $n$.

Step 4 (a common threshold). For $U\in\mathcal{U}$ let

$$W_U=\{N\in\mathbb{N}:\ x_m\notin U\ \text{for every}\ m\in\mathbb{N}\ \text{with}\ N\le m\},$$

which is nonempty by the definition of $\mathcal{U}$, and set $N_U=\min W_U$, which exists by The Natural Numbers Are Well Ordered. Since $N_U$ is determined by $U$, this defines the tuple $c$ in $\mathbb{N}$ with components $c_i=N_{\beta(i)}$ for $i\in[n]$, with no appeal to a choice principle. By claims 1, 2, and 3 of Properties of the Order on the Natural Numbers, the order $\le$ on $\mathbb{N}$ is reflexive, transitive, antisymmetric, and any two natural numbers are comparable, so it is a total order. Hence Greatest Element of a Finite Family in a Totally Ordered Set provides $j\in[n]$ with $c_i\le c_j$ for every $i\in[n]$. Put $N=c_j$.

Step 5 (contradiction). Consider the index $m=N$, which satisfies $N\le m$ by reflexivity of $\le$. By hypothesis $x_N\in K$, so by Step 3 there is $U\in J$ with $x_N\in U$. Since $\beta$ is a bijection onto $J$, there is $i\in[n]$ with $U=\beta(i)$, and then $N_U=c_i\le c_j=N$ by Step 4. As $N_U\in W_U$, the defining property of $W_U$ applied with the index $N$, which satisfies $N_U\le N$, gives $x_N\notin U$. This is a contradiction.

Therefore the assumption was false, and some $x\in K$ is a cluster point of $(x_m)_{m\in\mathbb{N}}$ in $(X,d)$.