I have been doing some conversions of fractional binary numbers to decimals, and I noticed a pattern: the decimal expansion always had $\le$ digits than the binary expansion. For example: $$\begin{align*} (0.1)_2 & = (0.5)_{10}\\ (0.01)_2 & = (0.25)_{10}\\ (0.001)_2 & = (0.125)_{10}\\ (0.1001101)_2 &= (0.6015625)_{10} \end{align*}$$ Does this pattern hold in general? I thought about this problem, and I was able to come up with a proof of the following weaker claim.
Claim: If $x \in \mathbb{R}$ has a finite binary representation, then it has a finite decimal representation.
Proof (Sketch): $x$ is a finite sum of numbers of the form $2^{-j}$. The set $$S = \{y \in \mathbb{R}| y \text{ has a finite decimal representation} \}$$ is closed under addition, so it suffices to prove that numbers of the form $2^{-j}$ have finite decimal representation. But $2^{-j}$ is just $1$ divided by $2$ repeatedly, and $S$ is closed under division by $2$, so $2^{-j}$ is in $S$ for every $j$.