Notation for applying a function recursively.

Question

Consider a function $f: \mathbb{R} \to \mathbb{R}$ that is monotonically increasing in $x$ such that $x_t = f(x_{t-1})$ where $t$ represents time.

Let $x_t \leq x'_t$, so $$ f(x_t) \leq f(f(x'_{t-1})) \leq f(f(f(x'_{t-2}))) \leq \cdots$$

Is there a more elegant way to write the expression with inequalities?

Note: I am unsure what tags to add.

Answer 1

$$f^n(x_{t-n+1}) \le f^{n+1}(x_{t-n}),$$ where the superscript indicates $n$-fold composition.