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?



Sequence that converge if they have an accumulation point
The following version of the mean ergodic theorem is taken from the book of Krengel, "ergodic theorems".


Let T be a bounded linear operator in a Banach space X. The Birkhoff averages are denoted by $A\_n = {1\over n} \ \Sigma\_{k=0}^{n-1} \ T^k$.
Assume that the sequence of operator norms $||A\_n||$ is bounded independently of $n$. Then for any x and y in B, the following is equivalent :


-- y is a weak cluster point of the sequence $(A\_nx)$,


-- y is the weak limit of the sequence $(A\_nx)$,


-- y is the strong limit of the sequence $(A\_nx)$.


(note that we talk about cluster points instead of converging subsequences because we didn't assume B separable. Hence the weak topology is not necessarily metrizable.)


This theorem implies e.g. the ergodic theorem for Markov operators on $C(K)$ (sequential compactness follows from Azrela-Ascoli), or the ergodic theorem for power bounded operators defined on reflexive Banach spaces (sequential compactness follows from Eberlein-Smulian). 


There is a whole set of theorems in ergodic theory along these lines. Let me mention the convergence of the one sided ergodic [Hilbert transform](http://www.ee.bgu.ac.il/~guycohen/CCL.pdf), discussed in Cohen and Cuny (see Th 3.2) as another example.


