prove that the sequence {(-1)^na with a subscript n} from n=1 to n=infinity diverges(has no limit)?

Question

if limit a with subscript n is not equal to zero.

Answer 1

I'll assume a_n is an integer for every n and you mean (-1)^(a_n).. (Othewise, (-1)^(a_n )may not be defined.

If this is the case, the statemant is not true. For example, if a_n = 2 for every n, the a_n converges to 2 and (-1)^(a)-n) = 1 for every n, so that it converges to 1.

Answer 2

I'm not sure that I've understood the comment about subscript n.

Is the term (-1)^(na)?

If it is, then (assuming a is an integer), the terms alternate between -1 and +1. A necessary condition for a sequence to converge is that there is some value of n, call it k, such that
absolute value of nth term < 1 for all n > k.

Since the absolute value of every term is equal to 1, this condition is not fulfilled.