Question: Let xn := n1/n for n N. (a) Show that xn+1 < xn if and only if (1 + 1/n)n < n, and infer

Let xn := n1/n for n ∈ N.
(a) Show that xn+1 < xn if and only if (1 + 1/n)n < n, and infer that the inequality is valid for n > 3. (See Example 3.3.6.) Conclude that (xn) is ultimately decreasing and that x := lim(xn) exists.
(b) Use the fact that the subsequence (x2n) also converges to x to conclude that x = 1.

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