Problem 198 (difficulty: 3/10)

Which of the following statements is equivalent to the negation of \(\displaystyle a_n\to A\)? What is the meaning of the rest? What are the implications among them?
 (1)  For all \(\displaystyle \varepsilon >0\) there are infinitely many members of \(\displaystyle a_n\) outside of \(\displaystyle (A-\varepsilon,A+\varepsilon)\).
 (2)  There is an \(\displaystyle \varepsilon >0\), such that there are infinitely many members of \(\displaystyle a_n\) outside of \(\displaystyle (A-\varepsilon,A+\varepsilon)\).
 (3)  For all \(\displaystyle \varepsilon >0\) there are only finitely many members of \(\displaystyle a_n\) in the interval \(\displaystyle (A-\varepsilon,A+\varepsilon)\).
 (4)  There is an \(\displaystyle \varepsilon >0\), such that there are only finitely many members of \(\displaystyle a_n\) in the interval \(\displaystyle (A-\varepsilon,A+\varepsilon)\).

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