Problem 207 (difficulty: 4/10)

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

\(\displaystyle (\forall P) (\exists n_0 ) (\forall n\ge n_0 )(a_n >P ).\)

Changing the quantifiers and the orders we can produce the following statements:
 (1)  \(\displaystyle (\forall P) (\exists n_0 ) (\exists n\ge n_0 )(a_n >P );\)
 (2)  \(\displaystyle (\forall P) (\forall n_0 ) (\forall n\ge n_0 )(a_n >P );\)
 (3)  \(\displaystyle (\exists P) (\exists n_0 ) (\forall n\ge n_0 )(a_n >P );\)
 (4)  \(\displaystyle (\exists P) (\exists n_0 ) (\exists n\ge n_0 )(a_n >P );\)
 (5)  \(\displaystyle (\exists n_0) (\forall P ) (\forall n\ge n_0 )(a_n >P );\)
 (6)  \(\displaystyle (\forall n_0) (\exists P ) (\exists n\ge n_0 )(a_n >P ).\) 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!