Prove that there are a countable number of intervals (in R) with rational endpoints
Counting Elements versus Counting Intervals
The interval [0,1) for example is uncountable, but we are not counting ELEMENTS of an interval, we are counting INTERVALS. [0,1] is one interval, [0,1/3] is another and so on.
Start with one end-point
Just count the left end points of [a,b]. Now for a fixed a there are countably many b’s which will form [a,b] as Q is countable. Now vary a∈Q. So there are countable union of countable elements.
Summary
There are a countable number of intervals (in R) with endpoints in the Rationals (Q).
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