Limit of a spiraling sequence in $\mathbb{R}^2$ I did some serious mistakes while typing a question few minutes ago.
Let $z_0=(0,0)$ and $z_1=(1,0)$ and define the sequence $z_n$, for any $ n \ge 2 $, as the endpoint of a line drawn perpendicular from $z_{n-1}$ and which is half the distance of the line joining $z_{n-2}$ and $z_{n-1}$
And drawn in a rectangular spiral fashion.
E.g., $z_2=(1,1/2), z_3=(3/4,1/2)$ and so on
Clearly convergent.
But What is the limit of $z_{n}$?

HINT
You can consider each coordinate separately. Look at the vectors of changes every 2 steps, so you have for $x$, for example:
$1, -1/4, 1/16, -1/64, \ldots$
which you must sum. It  is a geometric series with common factor $-1/4$ and initial value of $1$.
Similar picture will be in the $y$ direction.
