# 无限二等分[0,1]这个区间之后还剩下啥？what's left after dividing an unit interval [0,1] infinitely many times?

Dividing an unit interval
$[0,1]$
into two equal subintervals by the midpoint
$\dfrac {0+1} {2}=\dfrac {1} {2}$
, denote the left subinterval by
$I_{1}=\left[ 0,\dfrac {1} {2^{1}}\right]$
, next, divide
$I_{1}$
into two equal parts by its midpoint, denote the left subinterval by
$I_{2}=\left[ 0,\dfrac {1} {2^{2}}\right]$
. Keep repeating this procedure indefinitely, what's left in the end ? Since referred infinitely many times here, it seems impossible to image the end case, but we could actually 'see' it!

Continue the process, obtaining a sequence of nested intervals$
$I_{n}=\left[ 0,\dfrac {1} {2^{n}}\right], n = 1, 2, 3, ...$
$Applying the nested intervals theorem there is only one point, one real number 0 contained in every$
I_{n}
$, i.e.$
$\displaystyle\bigcap_{{n=1}}^{\infty}\left[ 0,\dfrac {1} {2^{n}}\right]=\left[ 0,0\right]=\{0\}$
$

In conclusion, the only thing left after infinitely many times of these dividing is a point. More general, we don't need to divide each interval equally to form that sequence of nested intervals, since the nested intervals theorem doesn't requires that.