Suppose there is a sequence xn=1/(n-2). We know we n tend to infinity the sequence tends to zero. But at n=2 it is equal to infinity. Is this sequence convergent? There is also a theorem that all convergent sequence are bounded for every n. But the sequence above is not bounded at n=2. From definition of convergent sequence it seems that only the case that n tends to infinity is concerned, it says nothing about whether it is convergent when n is finite but xn is not.