Let $S$ be the set of all infinite sequences of positive integers
$k_1, k_2,..., k_n,...$ which are strictly increasing.
The following "proof" of the statement that the cardinal number $|S| = c$ is incorrectly done by me(as my teacher told):
Proof : Identify each element of $S$ as a strictly increasing function $f : \mathbb{P} \to \mathbb{P}$ with $f(n) = k_n$.
For each $f\in S$, define
$$ a_n = 1 \ \text{if} \ n \in Im(f) \\ = 0 \ \text{if} \ n \notin Im(f)$$
for $n \in \mathbb{P}$.
This defines a function : $ \phi : S \to (0, 1]$ with
$\phi(f) = a_1a_2a_3...a_n...$
the binary representation of a real number in the interval $(0, 1] \subset \mathbb{R}$. Because it is a bijection, it follows that $|S| = |(0, 1]] = c$.
My question : help me find the error in the above incorrect "proof" and modify it, so that the same idea using the binary representation of decimal real numbers proves the statement.