Exercises: Set theoretic construction of real numbers

Exercises:

1. If p is a prime number, show that $\sqrt{p}$ is irrational.
2. Show that $\sum_{n=2}^{\infty} \frac{1}{n!}<1$
3. Show that if $e_{n} = \pm {1}$, then the number $\sim_{n=1}^{\infty} \frac{e_{n}}{n!}$ is an irrational number.
4. Show that if $\{ q_{n}\}$ and $\{ r_{n}\}$ are two Cauchy sequences, then so are $\{ q_{n}+r_{n}\}$and $\{ q_{n}.r_{n}\}$.
5. If A and B are non empty subsets of $\Re$ that is bounded below. Then, $s = \inf{A}$ if and only if (i) $s \leq a$ for all $a \in A$ and (ii) for any $\epsilon>0$, $A \bigcap [s, s +\epsilon) \neq \phi$.

Reference: A Second Course in Analysis by M Ram Murty, Hindustan Book Agency.

Regards,

Nalin Pithwa

This site uses Akismet to reduce spam. Learn how your comment data is processed.