Injectivity of the factorial-related map $f:\mathscr P(\Bbb N) \to \Bbb R$, $f(I) = \sum_{n \in I} \frac 1 {n!}$ Note: For the purpose of this question, $\Bbb N$ does not include $0$.
I have a function $f:\mathscr P(\Bbb N) \to \Bbb R$ defined by:
$$f(I) = \sum_{n \in I} \frac 1 {n!}$$
This is essentially a transformation from binary sequences indexed by $\Bbb N$ to a number in $\Bbb R$.
I would like to prove that this function is injective.

Hint First prove that $f(\{k\}) > f(\{k + 1, k + 2, \ldots\})$ for all $k \in \Bbb N$.
Then, consider distinct elements $I, J \in \mathscr{P}(\Bbb N)$ and the smallest $k \in \Bbb N$ which is in one and not the other. (You'll probably also want to use the apparent fact that $f$ is monotonic under inclusion, that is, that if $I \subseteq J$ then $f(I) \leq f(J)$.)
