if (an) is a sequence and (bn) is a sequence, then what is the sequence max(an, bn). Is it (an) if (an) converges to a higher limit and (bn) if (bn) converges to a higher limit?

also, can someone just tell me if the following statements are true or false? i'll prove them or come up with a counterexample, i just want to know which are true.

1.) if (an) and (bn) are divergent, then also (an + bn) and (an - bn) are divergent. I believe this is false.

2.) (an)^2 converges if and only if (abs(an)) converges. true?

3.) if (an) and (bn) are convergent, then also max(an, bn) is convergent. true?

4.) if (an+1 - an) converges to zero, then (an) converges. *the n+1 is all a subscript*

5.) if an > 0 and ((an+1)/(an)) < 1 for all n, then (an) converges. *again the n+1 is all a subscript*

I just want to know if they're true or false.