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MATH 4630‑01 Homework Assignment #2
DUE: Tuesday, Feb 17, 11:59p Sections 2.1–2.5
Instructions: Show all work clearly and justify each conclusion. Collaboration is encouraged, but write up your solutions individually in your own words.
For any prove/disprove problem: either give a proof, or give a specific counterexample (with a brief explanation of why it works).
  1. Using the definition of convergence, prove the following limits.
    1. $\displaystyle\lim_{n \to \infty}\left(\frac{3n+1}{2n+5}\right)=\frac{3}{2}$
    2. $\displaystyle\lim_{n \to \infty}\left(\frac{n}{n^2+1}\right)=0$
  2. Give an example of each or state that it is impossible.
    1. A sequence with an infinite number of ones that does not converge to one.
    2. A sequence with an infinite number of ones that converges to a limit not equal to one.
    3. A divergent sequence such that for every $n \in \mathbb{N}$ it is possible to find $n$ consecutive ones somewhere in the sequence.
  3. Let $(a_n) \to 0$. Use the Algebraic Limit Theorem to compute each limit (assuming fractions are defined):
    1. $\displaystyle\lim_{n\to \infty} \left(\frac{1+2a_n}{1+3a_n-4a_n^2}\right)$
    2. $\displaystyle\lim_{n\to \infty} \left(\frac{(a_n+2)^2-4}{a_n}\right)$
  4. Give an example of each, or state that it is impossible by referencing the appropriate theorem(s):
    1. Sequences $(x_n)$ and $(y_n)$ which both diverge, but whose sum $(x_n + y_n)$ converges.
    2. Sequences $(x_n)$ and $(y_n)$ where $(x_n)$ converges, $(y_n)$ diverges, and $(x_n+y_n)$ converges.
    3. A convergent sequence $(b_n)$ with $b_n \neq 0$ for all $n$ such that $(1/b_n)$ diverges.
    4. An unbounded sequence $(a_n)$ and a convergent sequence $(b_n)$ with $(a_n-b_n)$ bounded.
    5. Two sequences $(a_n)$ and $(b_n)$ where $(a_nb_n)$ and $(a_n)$ converge but $(b_n)$ does not.
  5. Prove that the sequence defined by $x_1:=1$ and $x_{n+1}:= \sqrt{2+x_n}$ for all $n \ge 1$ is convergent. Find the limit.
  6. Let $(a_n)$ be a bounded sequence.
    1. Prove that $y_n = \sup\{a_k : k \geq n\}$ converges.
    2. The limit superior is $\displaystyle\limsup_{n \to \infty} a_n = \lim_{n\to\infty} y_n$. Similarly define $\displaystyle\liminf_{n\to\infty} a_n = \lim_{n\to\infty} z_n$ where $z_n = \inf\{a_k:k\ge n\}$. Prove $(z_n)$ converges.
    3. Prove that $\displaystyle\liminf_{n \to \infty} a_n \leq \displaystyle\limsup_{n \to \infty} a_n$, and give an example where the inequality is strict.
    4. Show that $\displaystyle\liminf_{n \to \infty} a_n = \displaystyle\limsup_{n \to \infty} a_n$ if and only if $\lim_{n \to \infty} a_n$ exists.
  7. Give an example of each, or argue that such a request is impossible.
    1. A sequence that has a bounded subsequence but no convergent subsequence.
    2. A sequence not containing $0$ or $1$ as a term, but with subsequences converging to each of these values.
    3. A sequence with subsequences converging to every point in $\left\{1,\tfrac{1}{2},\tfrac{1}{3},\ldots\right\}$.
    4. A sequence with subsequences converging to every point in $\left\{1,\tfrac{1}{2},\tfrac{1}{3},\ldots\right\}$, and no subsequences converging to points outside this set.
  8. Decide whether the following propositions are true or false.
    1. If every proper subsequence of $(x_n)$ converges, then $(x_n)$ converges as well.
    2. If $(x_n)$ contains a divergent subsequence, then $(x_n)$ diverges.
    3. If $(x_n)$ is bounded and diverges, then there exist two subsequences converging to different limits.
    4. If $(x_n)$ is monotone and contains a convergent subsequence, then $(x_n)$ converges.
  9. Suppose that every subsequence of $(x_n)$ has a further subsequence converging to $0$. Prove that $\lim_{n \to \infty}x_n=0$. (Hint: use contradiction.)
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