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.

转载于:https://www.cnblogs.com/iMath/p/6397330.html