Question

Let S be proper subset of [0,1] consists of all finite decimal expansions x =0.a1a2a3....
where all but finitely many digits are 5 or 6.
Find sup S.
Please, help

Answer 1

I guess a_i can't all be zero and therefore can't all be 9
since 1=.99999999999999999999999999999999

Answer 2

so this set wouldn't include 0.0000000000000 or .999999999999999999999
or 1.0000000000000000000

Answer 3

digits are 5 or 6 only, not from 0 to 9

Answer 4

and a whole bunch of other numbers

Answer 5

oh

Answer 6

so , the first number in S , namely, x1 is
x1=0.5
x2 = 0.555555555555
xk= 0.5 555555555555556
x(k+1) =0.6
and so on....
I think so, but not sure

Answer 7

why isn't x2=0.55 ?
Or are you just making your own subset?
Also I don't know just asking...
also if the a_i's can only be 5 or 6 wouldn't that make the sup=2/3?

Answer 8

Honestly, I don't know how to make it up.

Answer 9

and i guess it couldn't be 2/3 because it says finite decimal expansions

Answer 10

The answer from back of the book is 1. But I don't know how.

Answer 11

i thought the sup was suppose to be included in the set

Answer 12

i didn't think 1 was in this set

Answer 13

:(

Answer 14

gtg, but appreciate what you leave. Please, let your guidance here, I 'll take it later. I am so sorry for not being here to get help.

Answer 15

ok maybe the sup number doesn't have to be included in the set

Answer 16

ok

Answer 17

I wonder if we say something like since the sup(0,1) =1
then any subset of (0,1) will also have the sup=1.
I wonder if there is some kinda theorem that proves that.

Answer 18

infinite subset of (0,1)*