This is in reference to Question 16 which is referring to 1.44 and 1.46 section of Walter Rudin's Functional Analysis. Please help me in starting this problem.
Below is the reference of the book pdf
https://59clc.files.wordpress.com/2012/08/functional-analysis-_-rudin-2th.pdf
If $\{H_n\}$is another sequence of compact sets satisfying the same properties then $H_n \subset \cup K_m^{0}$ because the interiors of the sets $K_m$ cover $\Omega$. By compactness, $H_n$ is contained in some $K_m$. Hence uniform convergence of a sequence of functions on each $K_m$ implies uniform convergence of the sequence on each $H_n$ and conversely.