Homework 5: The Limit of a Sequence

Instructions. Assignments should be stapled and written clearly and legibly.

1. Section 4.1, #6(c)(d), 14.
2. Using only the definition of a limit, prove
   1. \( \lim\limits_{n\to \infty} \dfrac{n^2-2}{n^4+2}=0\)
   2. \( \lim\limits_{n\to \infty}\, \dfrac{2n-2}{5n+3}=\dfrac{2}{5}\)
   3. \( \lim\limits_{n\to\infty}\, \Bigg(5-\dfrac{1}{\sqrt{n+\sqrt{n}+12}}\Bigg)=5\)
3. Consider the following definition: A sequence \((a_n)\) is said to reverge to \(L\) if there exists \(\epsilon>0\) such that for every \(N\in\mathbb{N}\), whenever \(n\geq N\), \(|a_n-L|\lt \epsilon\).
   1. Give an example of a sequence that reverges.
   2. If a sequence reverges, must it also converge? If not, give a counterexample.
   3. Is it possible for a sequence to reverge to two different values?
   4. (optional) If possible, give a simpler definition of revergence.
4. Suppose that a sequence \((a_n)\) converges to 0, and that at least one term \(a_n\) of the sequence is greater than \(0\). Prove that the set \(\{a_n\}\) has a maximum.