Decide whether the following propositions are true or false, providing a short justification or proof 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 monotomne and contains a convergent subsequence, then (xn) converges.
(e) If (xn) and (yn) are strictly increasing, then (xn+yn) is strictly increasing.
(f) If (xn) and (yn) are strictly increasing, then (xnyn) is strictly increasing.
(g) The sequence (xn) where x₁=.1,x₂=.101,x₃=.101001,x₄=.1010010001,dots and so on converges.
