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feat(theories/analysis): add weak squeeze theorem for converges_to_at

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Rob Lewis 2016-02-17 13:48:19 -05:00 committed by Leonardo de Moura
parent 56ca41a916
commit c87e79ff7f
1 changed files with 19 additions and 1 deletions
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20
library/theories/analysis/real_limit.lean
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@ -402,7 +402,7 @@ end limit_operations
/- properties of converges_to_at -/
section limit_operations_continuous
variables {f g : ℝ → ℝ}
variables {f g h : ℝ → ℝ}
variables {a b x y : ℝ}
theorem mul_converges_to_at (Hf : f ⟶ a [at x]) (Hg : g ⟶ b [at x]) : (λ z, f z * g z) ⟶ a * b [at x] :=
@ -594,3 +594,21 @@ theorem continuous_mul_of_continuous {f g : ℝ → ℝ} (Hconf : continuous f)
end
end continuous
-- this can be strengthened: Hle and Hge only need to hold around x
theorem converges_to_at_squeeze {M : Type} [Hm : metric_space M] {f g h : M → ℝ} {a : ℝ} {x : M}
(Hf : f ⟶ a at x) (Hh : h ⟶ a at x) (Hle : ∀ y : M, f y ≤ g y)
(Hge : ∀ y : M, g y ≤ h y) : g ⟶ a at x :=
begin
apply converges_to_at_of_all_conv_seqs,
intro X HX,
apply converges_to_seq_squeeze,
apply all_conv_seqs_of_converges_to_at Hf,
apply HX,
apply all_conv_seqs_of_converges_to_at Hh,
apply HX,
intro,
apply Hle,
intro,
apply Hge
end