Theorem climi0

Description: Convergence of a sequence of complex numbers to zero. (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 )
	climi0.5	⊢ ( 𝜑 → 𝐹 ⇝ 0 )
Assertion	climi0	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 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		climi0.5	⊢ ( 𝜑 → 𝐹 ⇝ 0 )
6	1 2 3 4 5	climi	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 0 ) ) < 𝐶 ) )
7		subid1	⊢ ( 𝐵 ∈ ℂ → ( 𝐵 − 0 ) = 𝐵 )
8	7	fveq2d	⊢ ( 𝐵 ∈ ℂ → ( abs ‘ ( 𝐵 − 0 ) ) = ( abs ‘ 𝐵 ) )
9	8	breq1d	⊢ ( 𝐵 ∈ ℂ → ( ( abs ‘ ( 𝐵 − 0 ) ) < 𝐶 ↔ ( abs ‘ 𝐵 ) < 𝐶 ) )
10	9	biimpa	⊢ ( ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 0 ) ) < 𝐶 ) → ( abs ‘ 𝐵 ) < 𝐶 )
11	10	ralimi	⊢ ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 0 ) ) < 𝐶 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝐶 )
12	11	reximi	⊢ ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐵 ∈ ℂ ∧ ( abs ‘ ( 𝐵 − 0 ) ) < 𝐶 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝐶 )
13	6 12	syl	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ 𝐵 ) < 𝐶 )