Let a, b ∈ R with 0 < a < b. Prove using the Density Theorem that there exists r ∈ Q such that r > 0 and a < sqrt(r)

Question

Density theorem: If x,y ∈ R and x<y then here exists r ∈ Q st. x<r<y

From the density theorem, it is clear that a<r<b. I am having trouble getting the sqrt(r). I do not really know where to begin with that.

HINT: pick any rational $r$ between $a^2$ and $b^2$. – Crostul Mar 03 '21 at 15:19 — Crostul, Mar 03 '21 at 15:19

score 1 · Answer 1 · answered Mar 03 '21 at 15:19

1

Hint: If $0<a<b$ then $a^2<b^2$. Apply the density theorem now and see what you get.

answered Mar 03 '21 at 15:19

ashK

1 Answers1