I'm referring to Problem 27 in the exercises of Chapter 2 in the textbook, Principles of Mathematical Analysis, 3rd edition, by Walter Rudin.
I've managed to prove that the set $P$ of condensation points of an uncountable set in $\mathbb R^k$ is closed, and now need to show that every point of $P$ is also a limit point.
A point $p$ in a metric space $X$ is siad to be a condensation point of a subset $E$ of $X$ iff every neighborhood of $p$ contains uncountably many points of $E$.
Where can I find the complete and accurate solutions manual for Rudin, and where can I find video lectures of a comprehensive course based on this book?
Best Answer
Consider a point $p$ in $P$.
Take a sequence of balls, $B_n$ of radius $\frac{1}{n}$ centered at $p$.
Because $p$ is a condensation point, for every $B_n$ there is a point $b_n$ in $P$, that is not $p$.
$b_n \to p$.