Bolzano–Weierstrass Theorem

Bolzano–Weierstrass Theorem: If $(x_n)$ is a bounded sequence in $\mathbb{R}^k$, then there exists a subsequence $(x_{n_j})$ and a point $x\in\mathbb{R}^k$ such that $x_{n_j}\to x \quad \text{as } j\to\infty.$

This theorem is the core compactness phenomenon in Euclidean spaces and underlies many existence proofs (maximizers/minimizers, convergence of approximations, etc.).

Proof sketch (optional): In $\mathbb{R}$, repeatedly bisect an interval containing the sequence to build nested intervals containing infinitely many terms and use nested intervals to extract a convergent subsequence. In $\mathbb{R}^k$, apply the one-dimensional result coordinatewise (diagonal subsequence argument).