Question
Given $a<b$, show that there are, in fact, infinitely many distinct rationals between $a$ and $b$. The same goes for irrationals, too.
Step 1
This means there is a positive distance between \( a \) and \( b \), which we can denote as \( d = b - a \). Show more…
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Consider $a, b \in \mathbb{R}$ where $a<b .$ Use Denseness of $\mathbb{Q} 4.7$ to show that there are infinitely many rationals between $a$ and $b.$
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.
Preliminaries
Real Numbers, Estimation, and Logic
Consider a, b ∈ R where a < b. Use the denseness of Q to show that there are infinitely many rationals between a and b.
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