Equivalents of the (Weak) Fan Theorem
... * Let (α)i be a sequence of real numbers (the lower bounds). Let (β)i be a sequence of real numbers (the upper bounds). If for all x ∈ [0, 1] there exists an i such that αi < x < β i, then there exists an m such that for all x ∈ [0, 1] there exists an i ≤ m such that αi < x < β i. ...
... * Let (α)i be a sequence of real numbers (the lower bounds). Let (β)i be a sequence of real numbers (the upper bounds). If for all x ∈ [0, 1] there exists an i such that αi < x < β i, then there exists an m such that for all x ∈ [0, 1] there exists an i ≤ m such that αi < x < β i. ...
