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.

Question

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.

Tristan Ward · Accepted Answer

Let   α be irrational. For each positive integer   n there is a least integer   k such that             k      n        &amp;gt;  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&#039;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  .

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

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