Suppose we split a line which length is $1$ in half. Then we get the $2$ lines which length is $1/2$ . Divide these two lines equally in half. Suppose we repeat this process infinitely. The length of lines can be calculated $\lim_{N \to \infty}$$\frac{1}{2^N}$ .But the sum of lines doesn't get $1$.
Let $\ \lim_{N \to \infty}$$\frac{1}{2^N}=\varepsilon$, where $\varepsilon \ge 0$
if $\varepsilon > 0$ then, $\sum_{n=1} ^{\infty} \varepsilon =\infty$
and if $\varepsilon = 0$ then, $\sum_{n=1} ^{\infty} \varepsilon =0$
Why does this contradiction occur?