Show that every irrational number in <mrow class="MJX-TeXAtom-ORD">

Nicholas Cruz 2022-05-23 Answered
Show that every irrational number in R is the limit of a sequence of rational numbers. Every rational number in R is the limit of a sequence of irrational numbers.
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Answers (1)

Tristan Ward
Answered 2022-05-24 Author has 8 answers
Let α be irrational. For each positive integer n there is a least integer k such that k n > α, and then this number is rational. You need to prove that this sequence tends to α.
Now let α be rational, and repeat the above argument with something like k 2 n instead of k n .
[You'll also need to prove that k 2 n is irrational whenever k 0, and cook up a way of avoiding having 0 in the sequence.] Edit: Or consider α + 2 n or something like that, as Cameron Buie suggests in the comments.
The moral of the story is that Q and R Q are dense in R .
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