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 = ( ZZ>= ` M )
	climi.2	|- ( ph -> M e. ZZ )
	climi.3	|- ( ph -> C e. RR+ )
	climi.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
	climi.5	|- ( ph -> F ~~> A )
Assertion	climi2	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < C )

Proof

Step	Hyp	Ref	Expression
1		climi.1	|- Z = ( ZZ>= ` M )
2		climi.2	|- ( ph -> M e. ZZ )
3		climi.3	|- ( ph -> C e. RR+ )
4		climi.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
5		climi.5	|- ( ph -> F ~~> A )
6	1 2 3 4 5	climi	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < C ) )
7		simpr	|- ( ( B e. CC /\ ( abs ` ( B - A ) ) < C ) -> ( abs ` ( B - A ) ) < C )
8	7	ralimi	|- ( A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < C ) -> A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < C )
9	8	reximi	|- ( E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < C ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < C )
10	6 9	syl	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < C )