- #1

kingwinner

- 1,270

- 0

_{n}), r

_{n}E (0,1), define a generalized Cantor set E by removing the middle r

_{1}fraction of an interval, then remove the middle r

_{2}fraction of the remaining 2 intervals, etc.

Start with [0,1]. Take r

_{n}=1/5

^{n}. Then the material removed at the n-th stage has length < 1/5

^{n}, so the total length removed is < 1/5 + 1/5

^{2}+ 1/5

^{3}+... = 1/4

Thus the length of E is >3/4. "

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I don't understand why the material removed at the n-th stage has length < 1/5

^{n}. How can we derive this? At the n-th stage, we are removing 2

^{n-1}pieces, so don't we have to multiply that by 2

^{n-1}?

I sat down and thought about this for half an hour, but I still can't figure it out.

I hope someone can explain this! Thank you!