The sequence [itex]a_n= (1)^n n[/itex] does NOT converge, even in the sense of "to [itex]\infty[/itex]" and "to [itex]\infty[/itex]". However, it has two subsequences to [itex]diverge[/itex] to those values: [itex]a_n[/itex], for n even, diverges to [itex]+\infty[/itex] while [itex]a_n[/itex], for n odd, diverges to [itex]\infty[/itex].
