I am looking for classes of sequence, that converge iff they contain a converging sub-sequence.

- The basic example of such sequences are monotone sequences of real numbers.
A more interesting examples comes from metric fixed point theory:

Let $B$ be a Banach space and $f\colon B \to B$ be a continuous mapping that is non-expansive (i.e. $\lVert f(x) - f(y)\rVert \le \lVert x -y\rVert$).

Define $x_{n+1} := \frac{1}{2} x_n + \frac{1}{2} f(x_n)$ for any startingpoint $x_0\in B$. This is the so called*Krasnoselski iteration*.

One can show that any accumulation point $\tilde{x}$ of $(x_n)$ is a fixed point of $f$. Since $f$ is non-expansive, it follows that

$\lVert x_{n+1}-\tilde{x}\rVert = \frac{1}{2}\lVert (x_{n}-\tilde{x}) + (f(x_{n}) - f(\tilde{x}))\rVert\le \lVert x_n -\tilde{x}\rVert$.

Hence $(x_n)$ converges iff it contains a converging sub-sequence.This is a special case of Ishikawa's fixed point theorem. (The Krasnoselski-Mann iteration - a generalization of the Krasnoselski iteration - also has this property.)

I am interested in this sequence because they provide very nice applications of the Bolzano-Weierstrass principle.

Do you know of any other examples of sequences with this property?

Do you know other proofs that uses this property together with the Bolzano-Weierstrass principle to prove the convergence of a sequence?