The Bolzano-Weierstrass Theorem is true in Rn as well: The Bolzano-Weierstrass Theorem: Every bounded sequence in Rn has a convergent subsequence. ... Proof: Every sequence in a closed and bounded subset is bounded, so it has a convergent subsequence, which converges to a point in the set, because the set is closed.

Do all sequences have a convergent subsequence?

Look at the definition! The nicest thing about these subsequences is a result attributed to the Czech mathematician and philosopher Bernard Bolzano (1781 to 1848) and the German mathematician Karl Weierstrass (1815 to 1897). Every bounded sequence has a convergent subsequence.

Can a divergent sequence have a convergent subsequence?

Furthermore, the Bolzano-Weierstrass Theorem says that every bounded sequence has a convergent subsequence. It depends on your definition of divergence: If you mean non-convergent, then the answer is yes; If you mean that the sequence "goes to infinity", than the answer is no.