Give an example of an infinite compact set $A$ such that its supremum is not a limit point

Question

I got this one on a quiz the other day (We're only working in the reals). My solution was

$$A=[0,1]\cup\{3\}$$

The closed interval has the infinite points, and $\sup A=3$ is not a limit-point since each neighborhood of $3$ contains only the point $3$.

I was marked wrong but I can't figure out why. Also let me know if there's any issues with my posting-- it's my first on here. Thanks for the input.

Answer 1

Another example is $$\{0\}\cup\biggl\{\frac1n: n\in\mathbb N\biggr\}$$ Maybe that was expected?

Answer 2

As stated, your answer appears correct, but I wonder what I would think if I saw all of what you wrote. You say each neighborhood of $3$ contains only $3$, and that's wrong: There is in fact only one open neighborhood of $3$ that contains only that point, and that neighborhood is $\{3\}$. The set $(1/2 - 1/100,1/2)\cup\{3\}$ is another open neighborhood of $3$, and it does not contain only that point.