Consider the closed interval

| August 30, 2017

1. Consider the closed interval [0,1]. Remove the open interval 11 I which has center

at 1/ 2and length 1/ 2 . 2 Then remove 12 I and 22 I from 11 I  I so that their centers

coincide with the center of closed interval of 11 [0,1] I and their length 1/ 2 . 4

Continuing in this way, construct 1/ 2 K in a similar way we constructed the Cantor

set K in class, as an intersection of all remaining closed intervals. And show that

length of the complement of 1/ 2 K is 1/ 2.

2.Use the induction to prove Bernoulli’s inequality: If 1+x>o, then (1+x)^n≥1+nx for all n in N

3.Let S be a nonenpty bounded subset of R and let m=sup S. Prove that m∈S iff m=max S

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