I am reading Principles of mathematical analysis by Walter Rudin and in have come across this theorem:
Theorem 2.41: If a set E in $R^k$ has one of the following three properties, then it has the other two:
- E is closed and bounded
- E is compact
- Every infinite subset of E has a limit point in E
My question is mainly about the third one. Why cannot I change it to "E contain all of its limit points"? why do I need to go through the trouble of defining it that way?
It seems that the original statement is stronger than my alternative one because if not it will be as same as saying compact sets and closed set are the same, which is not.
However, I do not see why is it stronger. Can you please clear out my confusion.
Best Answer
"$E$ contains all its limits point" is equivalent to "$E$ is closed" but not to "$E$ is closed and bounded".
For a counterexample : in $\mathbb R$, the set $E = \mathbb N$ of natural integers is closed but has no limit points. Therefore, it contains all its limit points (since there are none) but has an infinite subset without limit point in $E$.