(related to Proposition: \(b\)-Adic Fractions Are Real Cauchy Sequences)
It is sufficient to prove the proposition for a positive \(b\)-adic fraction.
According to its definition, a \(b\)-adic fraction is a real series, which means that it is a sequence \((s_m)_{m\in\mathbb N}\) of partial of sums \((s_m)_{m\in\mathbb N}\) with
\[s_m=\sum_{k=-n}^m a_kb^{-k}.\]
and \(0\le a_k < b\). We prove the proposition in two steps:
Let \(j\ge -n\) and let without loss of generality \(m \ge j\). Then we have
\[\begin{array}{rrl} |s_m - s_j|&=&\sum_{k=-n}^m a_kb^{-k} - \sum_{k=-n}^j a_kb^{-k}\\ &=&\sum_{k=j+1}^m a_kb^{-k}\\ &\le & \sum_{k=j+1}^m (b-1)b^{-k}\\ &=&(b-1)b^{-j-1}\sum_{k=0}^{m-j-1} b^{-k}\quad\quad ( * ) \end{array} \]
In the sum \( ( * ) \), we can apply the formula for the sum of geometric progression, namely
\[\sum_{k=0}^{m-j-1} b^{- k}=\sum_{k=0}^{m-j-1} \left(\frac 1b\right)^{k}=\frac{1-\left(\frac 1b\right)^{m-j}}{1- \frac 1b}=\frac{b-b^{j-m+1}}{b-1}.\]
By replacing this result in \( ( * ) \), we get
\[\begin{array}{rrl} |s_m - s_j|&=&(b - 1)b ^ { - j - 1}\cdot \frac{b - b^{j - m + 1}}{b - 1}\\ &=&b^{ - j } - b ^ { - m - 2}\\ &\le & b^{ - j} < \epsilon \end{array} \]
for any arbitrarily small \(\epsilon > 0\), if \(j\) gets sufficiently large. This proves that \((s_n)_{n\in\mathbb N}\) is a Cauchy sequence.
This follows immediately from the completeness principle of real numbers, due to which each real Cauchy sequence has a real limit.
