By the Bolzano–Weierstrass theorem we know that a bounded infinite sequence of real numbers has a convergent subsequence. What I'm wondering about is whether we can do more.
Here's my idea, which I'm not sure is correct: By applying Bolzano–Weierstrass again to the elements not participating in the convergent subsequence, we can get another convergent subsequence, and so on. We can always add to the new subsequence the first element of the original sequence not yet chosen for any subsequence, so this process generates a partition of the original sequence to an infinite number of convergent subsequences.
My question: can we partition a sequence into a finite number of convergent subsequences? My guess is no, but I couldn't find a counterexample.