I am new to analysis and dyadic expansion. One of the problem I came across is
If $\{x_n\}$, $\{y_n\}$ are two sequences of zeros and ones, show that
$\sum_{n=1}^{\infty}x_n/2^n = \sum_{n=1}^{\infty}y_n/2^n$
if and only if there is an integer $n$ such that $x_k=0$ and $y_k=1$ for all $k\geq n$.
I understand that every number has a unique binary expansion, but this problem just not intuitively make sense to me.
$$\sum_{n=1}^{\infty}x_n/2^n = \sum_{n=1}^{\infty}y_n/2^n$$ will not hold if you have all your $x_n =1$ and all you $y_n=0$ for $n\ge 1$
On the other hand if you have $x_1=0, x_n=1$ for $n\ge 2$ we will get
$$\sum_{n=1}^{\infty}x_n/2^n=1/2$$
And if we let $y_1=0$ and $y_n=1$ for all $n\ge 2$ we get $$ \sum_{n=1}^{\infty}y_n/2^n=1/2$$
Thus we have to modify the problem so it makes sense.