Can the closed unit interval be divided into smaller intervals?

[Jameson]

Here is this week's problem!

Show that the closed unit interval $[0,1]$ cannot be expressed into a disjoint union of closed intervals of length less than one.
----------------------------
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Euge]

No one answered this week's problem. You can find my solution below.

Spoiler

By way of contradiction, suppose $[0,1]$ can be written as a disjoint union of closed intervals of length less than one. These intervals form a countable collection, so we may enumerate them as $[a_n,b_n]$, $n\in \Bbb N$. Let $A = \{a_n : n\in \Bbb N\}$ and $B = \{b_n : n\in \Bbb N\}$. The set $S = (A \cup B)\setminus\{0,1\}$ is a compact set with no isolated point. This is a contradiction. For every compact metric space with no isolated point is uncountable.