Exercises 2.5 Exercises

Prove directly from the definition that \((1/n)\) is a null sequence.

Let \(x_n=1-1/n\text{.}\) Show that \((x_n)\) is increasing and converges. Find its limit.

Use the squeeze lemma to prove that \(\sin(n)/n\to 0\text{.}\)

Give a divergent sequence that has a convergent subsequence.

For each of the following sequences, compute its limit superior and limit inferior.

Show that the following statements are equivalent: (i) \(\vx \approx 0\text{,}\) (ii) \(-\vx \approx 0\) and (iii) \(|\vx| \approx 0\text{.}\)

Compute the limit superior and limit inferior, in the extended real numbers, of each of the following sequences.

Consider the sequence \(x_n = (-1)^n(1-1/n)\text{.}\)

Find a subsequence converging to \(\limsup x_n\) and a subsequence converging to \(\liminf x_n\text{.}\) Determine all limits of convergent subsequences of \((x_n)\text{.}\)

Suppose \(\sup S\) exists. Show that there is a sequence \((x_n)\) in \(S\) that converges to \(\sup S\text{.}\)

Let \((x_n)\) be a bounded sequence. Suppose every convergent subsequence of \((x_n)\) has the same limit \(L\text{.}\) Prove that \((x_n)\) converges to \(L\text{.}\)

Prove directly from the definition that \((1/n)\) is a Cauchy sequence.

Prove that every subsequence of a Cauchy sequence is Cauchy.

Give an example of a bounded sequence that is not Cauchy. Justify your answer.

Let \(x_n\) be the decimal truncation of \(\sqrt{2}\) to \(n\) decimal places, so that

Show that \((x_n)\) is Cauchy in \(\Q\text{.}\) Explain why it does not converge in \(\Q\text{,}\) and conclude that \(\Q\) is not sequentially complete.

Determine whether each sequence converges. If it does, find its limit.

Use the ratio test to prove that \(n^3/2^n\to 0\text{.}\)

Use the ratio test to prove that \(5^n/n!\to 0\text{.}\)

Give two sequences of nonzero real numbers \((x_n)\) and \((y_n)\) such that \(|x_{n+1}|/|x_n|\to 1\) and \(|y_{n+1}|/|y_n|\to 1\text{,}\) but \((x_n)\) is null while \((y_n)\) is unbounded.