Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn...

Let (x_n), (y_n) be bounded sequences.

a) Prove that lim inf x_n + lim inf y_n ≤ lim inf(x_n + y_n) ≤ lim sup(x_n + y_n) ≤ lim sup x_n + lim sup y_n. Give example where all inequalities are strict.

b)Let (z_n) be the sequence defined recursively by z₁ = z₂ = 1, z_n+2 = √ z_n+1 + √ z_n, n = 1, 2, . . . . Prove that (z_n) is convergent and find its limit. Hint; argue that 0 < z_n < 4 for all n and use part (a).

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Colby Messinger answered 1 year ago

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Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn...

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