Why would you think there is a largest?

Here's a short proof there isn't a largest number less than 1:

Assume that there is a largest number less than 1. Let's represent this number by [tex]\inline{x}[/tex].

Now, consider the number [tex]\inline{y=(1+x)/2}[/tex]

First, notice that [tex]\inline{y>x}[/tex]:

[tex]
y = \frac{1+x}{2} > \frac{x+x}{2} = x
[/tex]

Now, notice that [tex]\inline{y<1}[/tex]:

[tex]
y = \frac{1+x}{2} < \frac{1+1}{2} = 1
[/tex]

So consider carefully what we have proven:

If we assume there is a largest number less than 1, we can find a number less than 1, yet larger than the largest number less than 1.

That is a contradiction; our assumption that there is a largest number less than 1 must be false.

#geegn

9 comments recommend bookmark subscribe