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

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

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