I am self studying real analysis using Rudin "Principles of Mathematical Analysis". I have come up with the proof for one of the theorems, which is slightly different to Rudin one. Can you please help me to confirm or disprove it? Thanks a lot!
Theorem: $E$ is infinite subset of a compact set $K$, then $E$ has a limit point in $K$.
Proof:
Assume $\forall x \in K$, $x$ is not limit point in $E$. Then $\forall x \in K$, $\exists$ neighborhood $V(x)$ of $x$ which includes up to one member of $E$.
But $K$ is compact so $\exists$ an open cover of $K$ such that $K \subset \cup_{i=1}^{n}V(x_i)$, where $n$ is natural number. This cover contains at most $n$ members of $E$ and hence can not be an open cover for $E$, which causes contradiction, since $E \subset K$.