intuition.
destruct (eq_dec a zero).
trivial.
exfalso.
specialize (H (half a)).
apply f.
eapply le_half.
apply H.
eapply half_nz.
trivial.
Qed.
Definition left_total(r : nat -> A -> Prop) :=
forall n, exists a, r n a.
Definition functional(r : nat -> A -> Prop) :=
forall n a1 a2,
r n a1 ->
r n a2 ->
eq a1 a2.
Definition inf_limit(f : nat -> A -> Prop)(a : A) :=
forall (epsilon : A),
~eq epsilon zero ->
exists n : nat,
forall (n' : nat),
n' >= n ->
forall a',
f n' a' ->
le (distance a' a) epsilon.
(* Note: if the codomain of the relation is empty (after any point), then we can assign any limit to it. *)
(* We could define inf_limit in terms of inf_limit_2, but inf_limit_2 only shows up in proofs, while inf_limit shows up in top-level definitions. *)
Definition inf_limit_2(f f' : nat -> A -> Prop) :=
forall (epsilon : A),
~eq epsilon zero ->
exists n : nat,
forall (n' : nat),
n' >= n ->
forall a a',
f n' a ->
f' n' a' ->
le (distance a a') epsilon.

Theorem inf_limit_2_const : forall (f : nat -> A -> Prop)(a : A),
inf_limit_2 f (fun x => eq a) <->
inf_limit f a.