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#### bw0young0math

##### New member

- Jun 14, 2013

- 27

"For arbitrary irrational number a>0, let A={n+ma｜n,m are integer.}

Show that set A is dense in R(real number)

- Thread starter bw0young0math
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- Thread starter
- #1

- Jun 14, 2013

- 27

"For arbitrary irrational number a>0, let A={n+ma｜n,m are integer.}

Show that set A is dense in R(real number)

- Jan 30, 2012

- 2,502

Prove by contradiction that the smallest positive linear combination of any two real numbers divides both numbers. Deduce that the set of positive linear combinations of $a\in\mathbb{R}\setminus\mathbb{Q}$ and 1 does not have the smallest element (otherwise, $a$ and 1 would be commensurate). Next show that the greatest lower bound of the set of positive linear combinations is 0. Now that you have a positive linear combination as small as you'd like, note that $A$ contains all its multiples.