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(** * Application : construction of PI *)
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Definition PI_tg (n:nat) := / INR (2 * n + 1).

(* Specifications: *)
Lemma PI_tg_pos : forall n:nat, 0 <= PI_tg n.
Lemma PI_tg_decreasing : Un_decreasing PI_tg.
Lemma PI_tg_cv : Un_cv PI_tg 0.
Lemma exist_PI : { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt PI_tg) N) l }.