Consider the set of divergent sequences:
Is this a vector space? Explain.
I think this is not a vector space because if the limit of the sequence is DNE, then it is not in R. Is that correct?
Except that that last statement is untrue.
If it were true that those are divergent sequences whose sum was also a divergent sequence, that would prove nothing at all. In particular, it would not prove that "the sum of any/ two divergent sequences. Plato suggested you look at those two sequences because they give a counter-example. In particular, is NOT divergent. Actually write out four or five terms of the sequence and look at the sum .