If is a convergent subsequence it means the the terms of this sequence have to get sufficiently close to eachother far enough into the sequence (this is Cauchy's criterion). The terms of the full sequence consist of only it means if is a convergent subsequence then it must be eventually constant. Therefore, the only possible subsequential limits are . The limsup is the largest of these subsequential limits and the liminf is the smallest of these subsequential limits.