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Let $B:=\{(b_1, b_2, b_3, \ldots) : b_i =\pm i!$ for every $i \in \mathbb{N}\}$.

I WTS that $B$ is uncountable. I know there are several ways to do this. At this point I think that constructing a surjective function from $B$ into $[0,1]$ is the easiest way. However, this should require using decimal (maybe binary) expansions of the reals and I'm unsure of how to do work with these formally.

Therefore, it would be helpful to see what such a surjection would look like and why it works.

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    $\begingroup$ Is the ! (factorial) a part of this? $\endgroup$
    – Q the Platypus
    Commented Feb 27, 2017 at 2:56
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    $\begingroup$ Do you already know that the set of infinite binary sequences is uncountable? That's a much easier thing to project this onto . . . $\endgroup$
    – Noah Schweber
    Commented Feb 27, 2017 at 2:57
  • $\begingroup$ Also it might be easier for you to create an injective function from [0,1] into B. $\endgroup$
    – Q the Platypus
    Commented Feb 27, 2017 at 2:57
  • $\begingroup$ @NoahSchweber Yes I can use that fact. $\endgroup$
    – CuriousKid7
    Commented Feb 27, 2017 at 3:01
  • $\begingroup$ @QthePlatypus I hadn't thought of that, but if that's easier I'd be interested to see that instead. Also, yes the factorial is part of it as described. $\endgroup$
    – CuriousKid7
    Commented Feb 27, 2017 at 3:02

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