Need help with number 4, I have no idea how to do this problem. This is real analysis by the way. The text appears to be from a mathematics textbook, focusing on sequences and limits. Below is a transcription tailored for an educational website:---**1.3.19 Corollary**: Let \(\{a_n\}\) be a sequence in \(\mathbb{R}\).(a) If \(a \in \mathbb{R}\) such that \(\limsup a_n > a\), then there are infinitely many values of \(n\) such that \(a_n > a\).(b) If \(a \in \mathbb{R}\) such that \(\limsup a_n < a\), then there is an integer \(N\) such that \(a_n < a\) for all \(n \geq N\).**Proof**:(a) By the preceding proposition there is a subsequence \(\{a_{n_k}\}\) such that \(a_{n_k} \to \limsup a_n > a\). Thus there is an \(N\) such that \(a_{n_k} > a\) for all \(n_k \geq N\).(b) If \(\limsup a_n < a\), then the definition of the lim sup implies there is an \(N\) with \(\sup\{a_n, a_{n+1}, \dots\} < a\). Hence part (b).---**Exercises**:1. Prove Proposition 1.3.3(b).2. For each of the following sequences \(\{a_n\}\) find the value of the limit and for each stipulated value of \(\varepsilon\), find a value of \(N\) such that \(|a_n - a| < \varepsilon\) when \(n \geq N\). (a) \(a_n = n^{-1}\), \(\varepsilon = .0001\), \(a_n = 2^{-n}\), \(\varepsilon = .0001\). (Are the values for \(N\) you found the smallest possible? This has no bearing on the convergence, but it's a bit more challenging to find the smallest possible \(N\).)3. If \(a_n \to a\) and \(\{a_{n_k}\}\) is a renumbering of the original sequence, does \(a_{n_k}\) converge to \(a

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a E R; when a = too, we leave the proof to the reader. Let Sn suplan, an+1,...}; so Sna. By the definition of the limit superior, for each k ≥ 1 there is an mk with a ≤ Smk a, then there are infinitely many values of n such that an> a. = (b) If a € R such that lim sup an a. Thus there is an N such that ank > a for all nk ≥ N. ank (b) If lim sup an

a E R; when a = too, we leave the proof to the reader. Let Sn suplan, an+1,...}; so Sna. By the definition of the limit superior, for each k ≥ 1 there is an mk with a ≤ Smk a, then there are infinitely many values of n such that an> a. = (b) If a € R such that lim sup an a. Thus there is an N such that ank > a for all nk ≥ N. ank (b) If lim sup an

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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