In other words, can $\mathsf{RCA}_0$ prove that for every function $f\colon \mathbb{N} \to \mathbb{N}$, there is $b \in \mathbb{N}$ such that

$$ \exists k \in \mathbb{N},\ f(k) = b\quad \land\quad \forall j \in \mathbb{N},\ f(j) \geq b $$

If so, is there a reference to this result? If not, is this result known to be provable/equivalent over $\mathsf{RCA}_0$ to another subsystem of SOA, and is there a reference for this?

More generally (this is more than I need, but out of interest), we could consider the principles $\mathsf{L}\Sigma^i_n$, $\mathsf{L}\Pi^i_n$, $\mathsf{L}\Delta^i_n$ which say that every $\Sigma^i_n$ / $\Pi^i_n$ / $\Delta^i_n$ definable subset of $\mathbb{N}$ has a least element. Where do these fall in the reverse mathematical hierarchy? Is anything known (even for $i=0$)?

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