The problem is to prove that any set of disjoint non-overlapping intervals that contain more than a single point ("non-singular intervals) is countable. The text seems to want me to offer the following "proof:"
$1$. Start with the first interval and associate it with the number one.
$2.$ For each interval, assign its successor to $n+1$
$3.$ This is an injection from the set of intervals to the set of natural numbers, so the set is countable.
The problem I have with this "proof" is that is that for any interval in the set, there is no guarantee that it has a unique predecessor. Consider the union of two sets of intervals
First Set :
$(1/(n+1),1/(n+2))$ for $n$ all natural numbers $n$
Second set :
An unspecified set of intervals to the right.
The first interval in the second set has have no unique predecessor since there is no maximum value for $n$
I thought about trying to show that if an infinite set of intervals contained in $(0,1)$ were uncountable then it would have to include at least one singular interval.
My difficulty with thinking about this approach is that I don't have any examples of uncountable sets other than the reals, the rationals, etc. I would like to construct an uncountable set of intervals as an example.
I will be asking on meta how to embed markdown and LaTex in my next post.