The lemma is nice but non-essential to what follows. However, the next result is very illuminating. If a sequence has a unique point of accumulation, then it converges to that point. If a sequence converges, then it is bounded. As m was a minimum, M is a maximum.
Every bounded sequence has an accumulation point.
Hence, t is a point of accumulation for the sequence. Two sets have the same cardinality if there is a bijection from one onto the other. A set is finite if it is the empty set or if it can be obtained by adding one element to a finite set. The cardinality of a finite set is 0 if it is empty and one more than the cardinality of any finite set from which it is obtained. Thus, cardinality of a finite set is a nonnegative integer.