Largest x, x^2 less than 2

Date: 10/23/1999 at 08:16:58
From: Michael
Subject: Logic

Dear Dr. Math,

Could you please assist me with the following question:

Prove that there is no largest real number x, such that x^2 < 2. (Use 
indirect proof.)

Thanking you in advance,
Michael

Date: 10/25/1999 at 02:22:35
From: Doctor Floor
Subject: Re: Logic

Hi, Michael,

Thanks for your question.

Suppose that x is the largest x such that x^2 < 2.

     Then x < sqrt(2)

Let y = [x+sqrt(2)]/2

     then x < y < sqrt(2)

     and  y^2 < 2

And we have a contradiction.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/