Workshop Exercises 2.

1. For any $n>1$, let $a_n=\frac{1}{{\log }_2(n)}$ (also, let $a_1=1$). Show that $a_n\to 0$, by finding for any $\epsilon >0$ an explicite value of $N_{\epsilon }$ such that if $n\ge N_{\epsilon }$ then $a_n\le \epsilon$.

2. Give an example of two non-convergent sequences of positive real numbers ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$ and ${\{b_n\}}_{n=1}^{\mathrm{\infty }}$ so that the sequence ${\{a_n{b}_n\}}_{n=1}^{\mathrm{\infty }}$ is convergent.

3. Somebody came up with the definition of a Dauchy-sequence: A sequence of real numbers ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$ is a Dauchy-sequence if

   $\exists \epsilon >0\exists N>0$ such that if $n,m\ge N$ then $|a_n-a_m|\le \epsilon$.

   • Show that all Cauchy-sequences are Dauchy, but some Dauchy-sequences are not Cauchy-sequences.

   • Show that all Dauchy-sequences are bounded.

   • Show that all bounded sequences are, in fact, Dauchy-sequences.

4. In this problem we have infinitely many sequences:

   The first sequence is: ${\{x_n^1\}}_{n=1}^{\mathrm{\infty }}$. The second sequence is ${\{x_n^2\}}_{n=1}^{\mathrm{\infty }}$ and the $1000$-th sequence is ${\{x_n^{1000}\}}_{n=1}^{\mathrm{\infty }}$. So, in general, the $k$-th sequence is ${\{x_n^k\}}_{n=1}^{\mathrm{\infty }}$.

   • Is it true that if ALL the sequences above are bounded, then the sequence ${\{x_k^k\}}_{k=1}^{\mathrm{\infty }}$ is bounded as well?

   • Is it true that if ALL the sequences above are converging to $0$, then the sequence ${\{x_k^k\}}_{k=1}^{\mathrm{\infty }}$ is converging to $0$ as well?

   • Suppose that the following statement holds:

     For any $k\ge 1$ and $n\ge 1$: .

     Show that $x_k^k\to 0$.

   • (somewhat harder problem) Suppose that ALL the sequences are converging to $0$. Show that for any $k\ge 1$, you can pick an element $x_{i_k}^k$ from the $k$-th sequence such that $x_{i_k}^k\to 0$.  (there is another problem on the next page!!!)

   • (a bit of a challenge) Let $f:\mathbb{N}\to \mathbb{N}$ a bijective map. These sorts of functions are called permutations of the natural numbers. Let ${lim}_{n\to \mathrm{\infty }}{a}_n=a$. Show that ${lim}_{n\to \mathrm{\infty }}{a}_{f(n)}=a$. The sequence ${\{a_{f(n)}\}}_{n=1}^{\mathrm{\infty }}$ is called a rearrangement of the sequence ${\{a_n\}}_{n=1}^{\mathrm{\infty }}$.