Metamath Proof Explorer


Theorem xlimclim

Description: Given a sequence of reals, it converges to a real number A w.r.t. the standard topology on the reals, if and only if it converges to A w.r.t. to the standard topology on the extended reals (see climreeq ). (Contributed by Glauco Siliprandi, 5-Feb-2022)

Ref Expression
Hypotheses xlimclim.m ( 𝜑𝑀 ∈ ℤ )
xlimclim.z 𝑍 = ( ℤ𝑀 )
xlimclim.f ( 𝜑𝐹 : 𝑍 ⟶ ℝ )
xlimclim.a ( 𝜑𝐴 ∈ ℝ )
Assertion xlimclim ( 𝜑 → ( 𝐹 ~~>* 𝐴𝐹𝐴 ) )

Proof

Step Hyp Ref Expression
1 xlimclim.m ( 𝜑𝑀 ∈ ℤ )
2 xlimclim.z 𝑍 = ( ℤ𝑀 )
3 xlimclim.f ( 𝜑𝐹 : 𝑍 ⟶ ℝ )
4 xlimclim.a ( 𝜑𝐴 ∈ ℝ )
5 df-xlim ~~>* = ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) )
6 5 breqi ( 𝐹 ~~>* 𝐴𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 )
7 6 a1i ( 𝜑 → ( 𝐹 ~~>* 𝐴𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 ) )
8 xrtgioo2 ( topGen ‘ ran (,) ) = ( ( ordTop ‘ ≤ ) ↾t ℝ )
9 reex ℝ ∈ V
10 9 a1i ( 𝜑 → ℝ ∈ V )
11 letop ( ordTop ‘ ≤ ) ∈ Top
12 11 a1i ( 𝜑 → ( ordTop ‘ ≤ ) ∈ Top )
13 8 2 10 12 4 1 3 lmss ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴𝐹 ( ⇝𝑡 ‘ ( topGen ‘ ran (,) ) ) 𝐴 ) )
14 eqid ( ⇝𝑡 ‘ ( topGen ‘ ran (,) ) ) = ( ⇝𝑡 ‘ ( topGen ‘ ran (,) ) )
15 14 2 1 3 climreeq ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( topGen ‘ ran (,) ) ) 𝐴𝐹𝐴 ) )
16 7 13 15 3bitrd ( 𝜑 → ( 𝐹 ~~>* 𝐴𝐹𝐴 ) )