Show that the set of irrational numbers fails to be closed

Layton Park

Layton Park

Answered question

2022-11-05

Show that the set of irrational numbers fails to be closed

Answer & Explanation

Marshall Flowers

Marshall Flowers

Beginner2022-11-06Added 20 answers

Notice that 1 2 n 2 is irrational for all n and that it converges to 0.
Kale Sampson

Kale Sampson

Beginner2022-11-07Added 6 answers

Based on what we already know, all real numbers are limits of irrationals. So select the rational (and therefore real) number q, let I be the set of irrational numbers and let a n be a sequence of irrationals converging to q. Since { a n } I is a sequence in I, with a limit outside of I, I is not closed by definition.

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