Every sequence has a subsequence converging to the lim sup of the original sequence, and a subsequence converging to the lim inf of the original sequence. What do you know about subsequences of convergent sequences?
It seems direct to me that since the lim sup is lim inf, then all sunseq converge to the same value. Thus unique accumulation point and hence the seq converges.. Would an explaination like this work?
For the other direction, write down the definitions, formally in terms of epsilons and deltas, for limit, lim sup, and lim inf, then just crank it out. I'm not going to give a full solution for both directions.