Question: a) Let (s n ) (0, +) be a sequence of real numbers. Prove that lim inf (1/s n ) = 1 / (lim sup

a) Let (s_n) (0, +) be a sequence of real numbers. Prove that

lim inf (1/s_n) = 1 / (lim sup s_n)

b) Let (a_n) (0, 100) be a sequence of real numbers, assuming that there are two subsequences (n_k) and (n'_k) of the sequence of natural numbers (n) ={1, 2, 3, . . . , }, such that, for all k N, we have that:

an inf{a_m; m > n_k} for all n > n_k.

Prove that (a_n) converges.

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