Let S is a subset of [0,1] consist of all infinite decimal expansions x =0.a1a2a3...... where all but finitely many digits are 5 or 6. Find sup S
Please, help

Question

Let S is a subset of [0,1] consist of all infinite decimal expansions x =0.a1a2a3...... where all but finitely many digits are 5 or 6. Find sup S
Please, help

loser66 · Answer

I know sup S = 1, but don't know how to put it in logic

anonymous · Answer

me neither but perhaps you could make a sequence like 
$$.95656...\
.995656...\
.9995656...$$ which follows the prescription and has a sup of one

loser66 · Answer

$[0,1]\supset S$ and 1 is sup ([0,1]), so that 1 is upper bound of S
Let $\varepsilon >0$  $x>1-\varepsilon$
$x = 0.999999.....(n times) ..5 > 1-10^n$

loser66 · Answer

@kirbykirby

kirbykirby · Answer

I'm kind of in the same boat as @satellite73 on this one. Since you can get as close to as 1 as you want with those sequences, then it makes sense for $\sup S =1$, but I'm not too sure how to formalize this one :\ I guess it would be somewhat analogous to a situation where you have $\sup\{ x \in \mathbb{R}\mid 0 < x < 1 \}=1$ But yeah, technically, you could specify some epsilon the way you did (though I think you mean $10^{-n}$ ;) ), and you can always find some $\varepsilon $ in that way as there exists $\alpha \in S$ such that $\alpha > 1-\varepsilon = 1-10^{-n} $ for any $n$ as large as you want it to be, and $\varepsilon$ will still be $> 0$

ikram002p · Answer

its okey to have sup of 1