Question
Show that between any two distinct real numbers therc arc infinitely many rationals and infinitely many irrationals.
Step 1
Step 1: Let's consider two distinct real numbers, a and b, with a < b. Show more…
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Show that between any two different real numbers there is a rational number. (Hint: If $a<b$, then $b-a>0$, so there is a natural number $n$ such that $1 / n<b-a$. Consider the set $\{k: k / n>b\}$ and use the fact that a set of integers that is bounded from below contains a least element.) Show that between any two different real numbers there are infinitely many rational numbers.
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Real Numbers, Estimation, and Logic
Prove that between every two rational numbers there is an irrational number.
The Foundations: Logic and Proofs
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