prove or disprove that if a sequence does not have a converging subsequence then the sequence tend to infinity how should I approach this? "Prove or disprove that if a sequence does not have a converging subsequence then the sequence tend to infinity or negative infinity."

The statement is not true.

For example the sequence $$\\{(-1)^nn\\}$$ does not tend to infinity and it does not have a convergent subsequence.

Try the statement for positive sequences and you may get a true proposition.