Notation question: Does $\inf$ require a subscript?

Question

I am wondering which is more acceptable notation to use.

If I am trying to denote the infimum of the set of numbers $\{\alpha_m:m\geq k, m,k \in \mathbb {N}\}$, which is more acceptable to write?

$$\inf\{\alpha_m:m\geq k, m,k \in \mathbb {N}\}$$ OR $$\inf_{m\geq k}\{\alpha_m:m,k \in \mathbb {N}\}$$

Or in summary, does the $\inf$ notation require a subscript?

Answer 1

Both notations are bad. Indeed, the condition $k \in \mathbb{N}$ is not relevant (and even misleading) inside the set $\{\alpha_m:m\geq k, m,k \in \mathbb {N}\}$. You could write instead:

For each $k \in \mathbb {N}$, let $x_k = \inf \{\alpha_m \mid m\geqslant k \}$. Or, let $E_k = \{\alpha_m \mid m\geqslant k\}$ and let $x_k = \inf E_k$.