Prove: If x and y are real numbers with x&lt;y, then there are infinitely many rational numbers in the interval [x,y]

Let x,y∈R such that x0 and byt archimedean property there exists a n∈N such that n(y−x)&gt;1. There exist a natural number m such that m⇐na →mn⇐xmn⇐x&lt;m+1na→mn⇐xmn⇐xmn⇐xmn⇐x&lt;m+1nm+1n∈(x,y) and m+1n is a rational number. Now if we consider the interval (x,m+1n)pr(m+1n,y) then by same argument there is a ration number in that interval and continuing this we have that there are infinitely many rational number in the interval (x,y). So there are infinitely many rational number in the interval [x,y]

Check the solution to "Prove: If x and y are real numbers..."

High School
Algebra
Pre-Algebra
Negative numbers and coordinate plane

Lipossig

Answered question

2021-01-02

Prove: If x and y are real numbers with x<y, then there are infinitely many rational numbers in the interval [x,y]

Answer & Explanation

cheekabooy

Skilled2021-01-03Added 83 answers

Let $x, y \in R$ such that x0 and byt archimedean property there exists a $n \in N$ such that $n (y - x) > 1$ . There exist a natural number m such that
$m \Leftarrow n a$

$\to \frac{m}{n} \Leftarrow x \frac{m}{n} \Leftarrow x < \frac{m + 1}{n} a \to \frac{m}{n} \Leftarrow x \frac{m}{n} \Leftarrow x \frac{m}{n} \Leftarrow x \frac{m}{n} \Leftarrow x < \frac{m + 1}{n} \frac{m + 1}{n} \in (x, y)$
and $\frac{m + 1}{n}$ is a rational number. Now if we consider the interval $(x, \frac{m + 1}{n}) p r (\frac{m + 1}{n}, y)$ then by same argument there is a ration number in that interval and continuing this we have that there are infinitely many rational number in the interval (x,y). So there are infinitely many rational number in the interval [x,y]

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Pre-Algebra

Didn't find what you were looking for?