Final answer:
The Bolzano-Weierstrass theorem states that every bounded sequence in the real numbers has a convergent subsequence. The given sequence x_n = 1 + 2(-1)^n is bounded and therefore has a convergent subsequence.
Step-by-step explanation:
The Bolzano-Weierstrass theorem states that every bounded sequence in the real numbers has a convergent subsequence. A sequence is bounded if there exists a number M such that every term of the sequence is less than or equal to M. In the given sequence x_n = 1 + 2(-1)^n, we can show that it is bounded by the values 1 and 3. Therefore, by the Bolzano-Weierstrass theorem, the sequence has a convergent subsequence.