Question

Show that a nonconvergent bounded sequence has two subsequences

each with a different limit, than is, (xn) is not convergent, but
its subsequence (yn) converges to L and (zn) converges to M where
L=/M

Answer 1

Every nonconvergent bounded sequence has at least one convergent subsequence due to the Bolzano-Weierstrass theorem. If the original sequence doesn't converge, it's possible to find two subsequences each converging to different limits.

Step-by-step explanation:

A nonconvergent bounded sequence in mathematics is a sequence of numbers that remains within a specific range (bounded) but does not approach a single value (nonconvergent). According to the Bolzano-Weierstrass theorem, every bounded sequence has a convergent subsequence. If (xn) does not converge, we can still find at least one subsequence (yn) that converges to a limit L. Since (xn) isn't convergent, not all of its subsequences can converge to L. Therefore, we must find another subsequence (zn) that converges to a different limit M where L ≠ M. This is possible due to the sequence's boundedness and lack of convergence, implying that it must oscillate and have different accumulation points, each of which can be the limit of a different subsequence.