Understanding Rudin’s proof of Ch 2 #17; the set of all numbers in $[0, 1]$ whose decimal expansion contains only $4, 7$ is closed

Question

The question is:

Let $E$ be the set of all $x \in [0,1]$ whose decimal expansion contains only the digits 4 and 7. Is $E$ compact?

There is a proof for this question here:

My question: How did we obtain that $|x-y|\ge 10^{-m}-\epsilon$ where $\epsilon \le \sum_{k=m+1}^{\infty}10^{-k}|x_k-y_k|$? I can see how why would hold if $x_m \ge 1$, but what about $x_m = 0$?

P.S. The linked question is quite dissimilar from mine.

Answer 1

$$x-y=\sum_{k=m}^\infty(x_k-y_k)10^{-k} =(x_m-y_m)10^{-m}+\sum_{k=m+1}^\infty(x_k-y_k)10^{-k}.$$ By the triangle inequality, $$\begin{align*} |x-y|&\ge|x_m-y_m|10^{-m}-\left|\sum_{k=m+1}^\infty(x_k-y_k)10^{-k}\right|\\ &\ge10^{-m}-\sum_{k=m+1}^\infty|x_k-y_k|10^{-k} \end{align*}$$ since $|x_m-y_m|\ge1$.