Convergent subsequences I'm studying for a real analysis exam tomorrow and I am curious about something:

Two true false questions:
a) Every sequence in the interval (0,1) has a convergent subsequence. 
b) Every sequence in the interval (0,1) has a subsequence that converges to a point in (0,1)

This dude on Yahoo Answers says both are false and uses the counterexample {1/n}. I think he is wrong because the sequence {1/n} isn't even in my interval! (It fails for 1). In fact I think both are true, and I don't know the proper reasoning but all I know is that I am not convinced by this supposed "counterexample"

Secondly, my book describes the following proposition: "Let the sequence {An} converge to the limit a. Then every subsequence of {An} also converges to the same limit a." In feel like this confirms the validity of (b).

Any input on the truth of these two would be very much appreciated.
For $(a)$, you get sequences that are bounded. Use the Bolzano-Weierstrass theorem, and the Monotone Convergence theorem.

For $(b)$, consider the sequence $a_n=\frac 1 {n+1}$. Then $a_n\in(0,1)$ but $\lim a_n=0$.

ADD You say


  Secondly, my book describes the following proposition: "Let the sequence $\{a_n\}$ converge to the limit $a$. Then every subsequence of $\{a_n\}$ also converges to the same limit $a$." In feel like this confirms the validity of $(b)$.


Note that that proposition talks about a convergent sequence, and the behaviour of its subsequences, while $(b)$ considers any sequence and it's subsequence, and the value it converges to. If you have an exam tomorrow, consider writing down the argument you "feel" confirms the validity of something, so that you know wether you're right or not, and reading propositions to understand what they are really stating. In this case, that proposition tells you  nothing about $(b)$, unless you consider the convergent sequences in $(0,1)$ that converge to a point in $(0,1)$. But then you're missing a lot of other sequences that don't converge, but are bounded, so have a convergent subsequence, which may, or may not converge to a point in $(0,1)$. In particular, this is because $0$ and $1$ are limit points of $(0,1)$. If we were talking about $[0,1]$, the answer would be affirmative, because $[0,1]$ us a closed subset of $\mathbb R$.
The dude is almost right. Both are false. Take $a_n = \frac{1}{2n}$.

By the way, are you sure the question is stated with open intervals?

Edit: I have misunderstood your question, my answer applies only if your topological space is just $(0,1)$.