Closed Subsets of Real Numbers

Could someone confirm my understanding of this definition?

A subset S of the real numbers is closed if there exists a sequence $$\left\{ a _{n} \right\} $$ in S such that if $$\lim_{n \rightarrow \infty}a _{n}=a$$ then $$a \in S$$

Is this correct?

Question

Closed Subsets of Real Numbers

Could someone confirm my understanding of this definition?

A subset S of the real numbers is closed if there exists a sequence $$\left\{ a _{n} \right\} $$ in S such that if $$\lim_{n \rightarrow \infty}a _{n}=a$$ then $$a \in S$$

Is this correct?

chriss · Answer

So basically if {a_n} converges to a, then a is in S.

chriss · Answer

This is the definition from the book.  I was attempting to reword it to my understanding?  Could you point to how what I said is not consistent with this definition?

A subset S of R (real numbers) is said to be closed provided that if {a_n} is a sequence in S that converges to a number a, then the limit a also belongs to S.

experimentx · Answer

sorry, i missed a point
{a_n} was sequence in S.

chriss · Answer

right, so my interpretation is correct then?

experimentx · Answer

i thought you were trying to construct a closed set based on convergence of sequence.

experimentx · Answer

yes, that's okay ... every closed set contains it's limit point. 'a' is a limit point since any open ball interval centered around 'a' contains infinitely many points of S.

chriss · Answer

awesome.  thanks so much!