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.