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We are given a metric space $(M,d(x,y))$ and a sequence $\{x_j\}_{j=1}^\infty$ of elements of a compact set $L\subseteq M$. Also K is the set of $x\in M$ for which there is a subsequence, $\{x_{j_n}\}_{n=1}^\infty$ that converges to $x$. If $K$ has exactly one element, then we have to show that $\{x_j\}_{j=1}^\infty$ converges to that element.

Is this problem the same as proving that if all possible subsequences of a given sequence of elements of a compact set converge to an element, then the sequence also converges to that same element?

And if not, how do we prove this result? By contradiction?

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up vote 2 down vote accepted

If $(x_j)$ doesn't converge to that single subsequential limit $x\in K$, then there is a subsequence $(y_j)$ of $(x_j)$ such that for some $\epsilon>0$, for all $j$, we have $d(y_j,x)\geq \epsilon$. Because $(y_j)$ is a sequence in the compact set $L$, it has a convergent subsequence, call it $(z_j)$, such that $(z_j)$ converges to some $y\in L$ where $y\neq x$. Since $(z_j)$ is a subsequence of $(x_j)$, we get $y\in K$, a contradiction.

Your saying that all possible subsequences of a given sequence of elements of a compact set converge to an element doesn't make sense here because that just means the sequence itself converges since it is a subsequence of itself.

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Okay, I accept your reasoning. Then how is the original problem to be solved? – schumi Dec 10 '12 at 3:14
oh! Damn, it's that simple! okay thanks a lot! – schumi Dec 10 '12 at 3:18
Yes, I was just doing that. I was starting by assuming that $x_j$ converges to some other point $y$ and following up on your logic. But is it safe to assume that the $x_j$ converges at all? What if it doesn't? – schumi Dec 10 '12 at 3:23

I don't think this is the same. That $K = \{x\}$ is a singleton means there exists a subsequence converging to $x$, but does not imply that every subsequence converges to $x$, since $\{x_j\}_{j=1}^\infty$ could have a nonconvergent subsequence in general. The question is asking: if every convergent subsequence of a sequence in a compact set converges to the same point $x$, then does the sequence itself converge to that point?

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Like Jasper says, if we take out convergent and assume every subsequence converges to $x$, then this is trivial since the sequence is a subsequence of itself. – Gyu Eun Lee Dec 10 '12 at 3:08
Okay I see your point. So how does one go about proving the original statement. – schumi Dec 10 '12 at 3:15

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