Real Analysis Prelim Camp

Here is where I will upload notes, problems, and solutions for the real analysis prelim camp in Summer, 2020. All the problems come from past UT prelim exams.

For problem 11, there is actually an alternative way using Borel-Cantelli. For each $\epsilon > 0$ and $n \in \mathbb{N}$ define the sets $A_n^\epsilon := \{x\in[0,1] : |f_n(x)| \geq \epsilon\}$ . Then Chebyshev implies that

$|A_n^\epsilon| \leq \frac{1}{\epsilon} \Vert f_n \Vert_{L^1([0,1])} \leq \epsilon^{-1} n^{-2}$

Then Borel-Cantelli implies that $\limsup_n A_n^\epsilon$ has measure zero. Now, define the set $A : = \bigcup_{k =1}^\infty A_n^{1/k}$ . Note that $A$ has measure zero and that $x \in A$ if and only if $f_n(x)$ doesn’t converge to zero. This shows that $f_n(x) \rightarrow 0$ for a.e. $x \in[0,1]$ .