Theorem climi2

Description: Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007) (Revised by Mario Carneiro, 31-Jan-2014)

	Ref	Expression
Hypotheses	climi.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	climi.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	climi.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
	climi.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
	climi.5	⊢ ( 𝜑 → 𝐹 ⇝ 𝐴 )
Assertion	climi2	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		climi.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2		climi.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		climi.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
4		climi.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = 𝐵 )
5		climi.5	⊢ ( 𝜑 → 𝐹 ⇝ 𝐴 )
6	1 2 3 4 5	climi	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) )
7		simpr	⊢ ( ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) → ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 )
8	7	ralimi	⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 )
9	8	reximi	⊢ ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 )
10	6 9	syl	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 )