Prove the following properties of sequences. Show your work as a mathematically rigorous argument. Remember this is a

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Prove the following properties of sequences. Show yourwork as a mathematically rigorous argument. Remember this isa writing-intensive course.
Problem 2.4
a) Let a_n be a sequence such that a_2n→L and a_2n+1 →L,then a_n→L.
b) Let a_n and b_n be sequences such that lim_n→∞ a_n= L ≠ 0 and lim_n→∞ a_nb_n exists, then lim_n→∞ b_nexists.
c) Prove that every unbounded sequence contains a monotonicsubsequence.