Well basically you need to show that every convergence sequence is bounded.
If convergence (say to ) then there is N such that for all we have
| .
Hence for . Now for every , so it is bounded.
A more direct way would warrant this. Let . So then for to converge to a finite value, let's say , we must have that . But now assuming that we are talking about we have that the defintion of a sequence converging to is for every there exists a such that . Now since are real there exists a number let say such that . So it follows that . Thus it is bounded.