Why convergence sequence must be bounded sequence? Don't go too far But I hope it makes me understand

Why convergence sequence must be bounded sequence? Don't go too far But I hope it makes me understand

Because the sequence converges, let, as defined by, for, there are positive integers,
n> When n, there are (n > n), so there are
Take, then for all N, there is, so the sequence is bounded
According to theorem 2, if a sequence is unbounded, it must be divergent. But it must be noted that a bounded sequence is not necessarily convergent. For example, a sequence is bounded, because it is divergent (see example 4). It can be seen that a sequence is bounded, which is a necessary condition for the convergence of a sequence, but not a sufficient condition

Sequence bounded is the same as sequence convergence. It's not the same. It's better to write down all the big ones
It is also said in the book that convergence sequence must be bounded,
I'm a freshman. This is senior mathematics. Please be simple, don't be too complicated
You can't tell me why.

1. The convergence of sequence must be bounded. It should be proved in the book. It is very simple that for any E > 0, there exists n > 0, such that for n > N, | an-c|

Boundedness and convergence of sequence
What is the condition of sequence convergence that the sequence is bounded?
A. Sufficient and necessary B. sufficient C. neither sufficient nor necessary D. necessary

D
The convergence sequence must be bounded
Let the sequence {an}, n > = 1 converge to a, then for any a > 0, there exists an n such that for all n > n there is an-a|