existing sequence and the bounded condition

Question

if there is a limit point exist on point E there guarantees the existence of a sequence {$Pn$}

can we say this sequence {Pn} is bounded?

Answer 1

Yes. The accumulation point can be viewed as the limit of a sequence. If a sequence converges, then it is bounded.

Answer 2

I'm not sure what your first sentence is supposed to be saying. Here are two suggestions, with the parenthetical portions added to give the second statement: "If a set $E$ of real numbers has a limit point, say $p$, then we are guaranteed the existence of a sequence $\{p_n\}$ of distinct points of $E$ (converging to $p$)."

In the first case we cannot necessarily conclude that $\{p_n\}$ is bounded. We can conclude the existence of a sequence, because $E$ will necessarily be infinite (it can't have a limit point, otherwise), but this sequence need not be bounded. For example, $\Bbb R$ has a limit point, and the sequence $p_n=n$ is a sequence of points of $\Bbb R,$ but the sequence is not bounded.

In the second case, the sequence is bounded, as all convergent sequences of real numbers are. All but finitely-many terms of the sequence lie in the interval $(p-1,p+1),$ and we can easily put a bound on the rest, since there are only finitely-many.