Let $ \ x \in \mathbb{R}^{\mathbb{N}} \ $ be an unbounded real sequence. I need to exhibit a detailed construction of a subsequence $ \ y = x \circ \sigma \ $ of $x$ such that $ \ y(n) > n \ $, for all $ \ n \in \mathbb{N} \ $. I am supposed to work on $\mathsf{ZF}$ system and I can use the axiom of countable choice. I cannot use dependent choice or full choice.
I tried to define $ \ \sigma : \mathbb{N} \to \mathbb{N} \ $ such that $ \ \sigma (n) = \min \big\{ m \in \mathbb{N} : x(m) > n \big\} $, but this is not necessarily strictly increasing and I need an strictly increasing $ \ \sigma \in \mathbb{N}^{\mathbb{N}}$ in order to define a subsequence.