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Lemmas about (finite domain) functions into fields. #

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We split this from algebra.order.field.basic to avoid importing the finiteness hierarchy there.

theorem pi.exists_forall_pos_add_lt {α : Type u_1} {ι : Type u_2} [linear_ordered_semifield α] [has_exists_add_of_le α] [finite ι] {x y : ι α} (h : (i : ι), x i < y i) :
(ε : α), 0 < ε (i : ι), x i + ε < y i