intros H.
induction H...
apply Rle_trans with (C m)...
rewrite (C_S (S m)).
simpl pred.
rewrite <- (Rplus_0_l (C m)) at 1.
apply Rplus_le_compat.
unfold Rdiv.
rewrite S_INR...
apply Rmult_le_pos...
apply Rmult_le_pos...
rewrite <- (Rmult_1_l (C m)) at 1.
apply Rmult_le_compat_r...
apply C_nonneg.
rewrite S_INR.
apply Rdiv_le_1...
Qed.
(* C is bounded by prefixes of the harmonic series: *)
Hint Resolve Rplus_le_le_0_compat.

Lemma C_bounded_by_harmonic n: C n <= n * 2 * Rsum (map (Rinv ∘ INR) (nats 2 (pred n))).
Proof with try field; auto with real.