Decide whether the following propositions are true or false, providing a short justification for each conclusion. (a) If every proper subsequence of (xn) converges, then (xn) converges as well. (b) If (xn) contains a divergent subsequence, then (xn) diverges. (c) If (xn) is bounded and diverges, then there exist two subsequences of (xn) that converge to different limits. (d) If (xn) is monotone and contains a convergent subsequence, then (xn) converges.