Can a subsequential limit be outside a bounded closed set?

[╔(σ_σ)╝]

Homework Statement

Please review the attachment.

Why can L be OUTSIDE of D ?

$$x_n \in D$$ and D is bounded. So doesn't that mean that any subsequential limit of $$x_n$$ has to be in D ? Btw, Theorem 3.10 is Bolzano Weierstrass.
Theorem 4.4 Says that A continuous function on a closed interval is uniformly continuous there.

 

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I think I understand now.

They said that D is bounded by not closed. i.e suppose D = (a,b)

$$x_{n_k} \to a$$ could happen and a in not in D.

Is this correct ? Anyone ?

 

[hgfalling]

What you said is right. Also, the bounded subset need not be an interval of reals; for example, the subset could be all the rational numbers from 0 to 2. Then there are an uncountable number of points not in the set that are the limit of Cauchy sequences inside the set.

 

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Perfect. Thanks.