Possible limits from the Bolzano-Weierstrass Theorem The Bolzano-Weierstrass Theorem states that

Every bounded sequence has a convergent subsequence.

For reference, I'll call the original bounded sequence $S_n$ and any subsequences $S'_k$.
I have a few questions about the limits of these subsequences.
Firstly, there are obviously infinitely many possible $S'$. Will these subsequences always have the same/different limits?
Secondly, I am guessing that there exists some $S'_k$ such that $\displaystyle \lim_{k \rightarrow \infty} S'_k = \lim_{n \rightarrow \infty} S_n$. If such a thing does exist, under what conditions will we have this?
Thirdly, in relation to my second question, is there a non-trivial $S_n$ such that any $S'_k$ always converges to the same limit as $S_n$?
 A: Every subsequence converges to the same limit if and only if the original series converges.
There are sequences in $[0,1]$ which have subsequences converging to all reals in $[0,1]$: take an enumeration of the rationals, for example.
There are also sequences which have subsequences converging to two reals; for example, $x_{2n} = \frac{3}{4}$ and $x_{2n+1} = \frac{1}{4}$.
So if the original sequence doesn't converge, then all bets are off without more specific knowledge of the sequence.
A: First off, you can have an uncountable set such that all its points are limits of subsequences of some original sequence. Here's a simple example:
Since the rational numbers are countable, let $S_n$ be a sequence listing all of the rational numbers in the interval $[0,1]$. Then, if you pick any real number $x$ in $[0,1]$, let $a_1$ be the first term in $S_n$ that is closer to $x$ than $2^{-1}$, let $a_2$ be the first term in $S_n$ (after $a_1$) that is closer to $x$ than $2^{-2}$, etc. This gives a subsequence $S^{\prime}_k$ of $S_n$ that converges to $x$.
On the other end of possible scenarios, if the original sequence is convergent, then the triangle inequality implies that every subsequence converges, and to the same limit. So, in answer to your second question, if you do have $\lim S^{\prime}_k=\lim S_n$, then all sub-sequences converge. 
However, not all bounded sequences converge. The sequence $S_n=(-1)^n$ is bounded, non-convergent, but the even terms converge to $1$ and the odd terms converge to $-1$.
If you get creative, you can construct sequences with behavior anywhere in between those two extremes (e.g. exactly 11 sub-sequential limit points, or countably many limit points, etc.).
The above show that the answer to the third question is yes, every convergent sequence has that property.
