Sequence that converge if they have an accumulation point I am looking for classes of sequence, that converge iff they contain a converging sub-sequence.


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*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?
 A: 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, discussed in Cohen and Cuny (see Th 3.2) as another example.
