Problem 206 (difficulty: 4/10)

Consider the definition of \(\displaystyle a_n \to b\):

\(\displaystyle (\forall \varepsilon >0) (\exists n_0 ) (\forall n\ge n_0 )(|a_n -b |<\varepsilon ).\)

Changing the quantifiers and their order we can produce the following statements:
 (1)  \(\displaystyle (\forall \varepsilon >0) (\exists n_0 ) (\exists n\ge n_0 )(|a_n -b |<\varepsilon );\)
 (2)  \(\displaystyle (\forall \varepsilon >0) (\forall n_0 ) (\forall n\ge n_0 )(|a_n -b |<\varepsilon );\)
 (3)  \(\displaystyle (\exists \varepsilon >0) (\exists n_0 ) (\exists n\ge n_0 )(|a_n -b |<\varepsilon );\)
 (4)  \(\displaystyle (\exists n_0) (\forall \varepsilon >0 ) (\forall n\ge n_0 )(|a_n -b |<\varepsilon );\)
 (5)  \(\displaystyle (\forall n_0) (\exists \varepsilon >0 ) (\exists n\ge n_0 )(|a_n -b |<\varepsilon ).\) Which properties of the sequence \(\displaystyle (a_n )\) are expressed by these statements? Give examples of sequences (if they exist) satisfying these properties.

Give me another random problem!