Convergence of a metric given by a series

Question

Let $\{{0, 1}\}^{\omega}$ be the set of all binary sequences, i.e., sequences of zeros and ones. It is pretty straightforward to verify that $$d(x, \, y) := \sum_{n\,=\,1}^{\infty} \frac{\mid{\,x_n - y_n}\mid\,}{2^{n}}$$ defines a metric in $\{{0, 1}\}^{\omega}$, but I'm not sure how to verify that $d(x,\,y) < \infty$. In other words, how can one bound the term $\mid{\,x_n - y_n}\mid$? My guess is that $\mid{\,x_n - y_n}\mid\,< 1$, but I can't write it properly. Any help is appreciated, thanks in advance!

Answer 1

$|x_n-y_n|$ can only have the values 1 or 0.

If $x_n=y_n$ it is zero.

If $x_n\ne y_n$ Then either $x=1$ and $y=0$ or the other way round. Either way $|x_n-y_n|=1$

As $0\le|x_n-y_n|\le1$, $$0\le d(x,y) \le \sum_1^\infty 1/2^n = 1$$ and indeed this bound is reached if the two sequences differ in every term. For example if $x= 1,1,1,\ldots$ and $y=0,0,0\ldots$.