Question

Prove that the irrational numbers are dense in r

Answer 1

(1) Prove that the irrational numbers are dense in R. In other words, show that if a, b ∈ R
and a < b, then there exists an irrational number x such that a < x < b.
(2) Suppose a, b ∈ R and a < b. Prove that the open interval (or “segment” in Rudin)
(a, b) = {c ∈ R : a < c < b}
has the same cardinality as R.
(3) Let S be a set of positive real numbers with the property that the sum of any finite
subset of S is always less than or equal to 1..

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