How to prove the following statement about non-terminating decimals?

Question

I was asked to prove the following statement:

Suppose $A=.a_1a_2a_3...$ and $B=.b_1b_2b_3..$ are non-terminating decimals in standard form with $A>B$. There is a rational number between $A$ and $B$.

Can somebody explain this to me? Thank you very much.

Answer 1

Find the minimum value of $n$ such that $a_n>b_n$.

Note that $\forall_{k<n}:a_n=b_n$.

Therefore $A=0.\overline{a_1\dots a_n\dots}>0.\overline{a_1\dots a_n}>0.\overline{b_1\dots b_n\dots}=B$.

Note that $0.\overline{a_1\dots a_n}=\overline{a_1\dots a_n}/10^n$.

Therefore $A>\overline{a_1\dots a_n}/10^n>B$.

Since both $\overline{a_1\dots a_n}$ and $10^n$ are integers, $\overline{a_1\dots a_n}/10^n$ is rational.