Problem:

Show that $\displaystyle S = \{x \in \mathbb{R} | 0 < x < 1\}$ has the same cardinality as $\displaystyle \mathbb{R}$.

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From my understanding (and using the book's definition), the cardinality is just the number of elements in the set. So I think I need to prove that there exists a one-to-one function mapping the infinitely many numbers between 0 and 1 to all real numbers $\displaystyle \mathbb{R}$.

I looked at cos(x) and sin(x) to maybe just try to say something like, function f(x) = cos(x) maps all real numbers into a number between 0 and 1, but this is not true. And even if I rigged that to work, it's not a one-to-one function since cos(x), sin(x) etc... can equal the same value with different inputs. Am I on the right track, though?

I don't want the answer, just a little guidance.

Thanks.