I was once given a question (in a take home exam) along the following lines: A sequence of complex-valued bounded measurable functions $f_n (x)$ on a probability space $X$ was given, with the functions $f_n$ satisfying some conditions which I won't specify here. The problem was to show that the $L^2 (X)$ limit of the sequence is a certain given function $f(x)$. After some thought, I came up with the following "solution": I defined a sequence of functions $g_n (x,y) \in L^2 (X \times X)$ and a function $g(x,y) \in L^2 (X \times X)$ such that $g_n (x,x) = f_n (x,x)$ x)$and$g(x,x)=f(x)$for all$x \in X$, and I proved that$g_n (x,y) \to g(x,y) \;$in$L^2 (X \times X)$. I was sure that I had, in fact, proved a generalization of the proposition I was given. Embarrassingly, it took me some time to realize my mistake. I tried to salvage my proof using continuity arguments and what not, but in the end I gave up and managed to concoct a different approach, which unfortunately was a lot more complicated and messy. 2 minor grammar fix I was once given a question (in a take home exam) along the following lines: A sequence of complex-valued bounded measurable functions$f_n (x)$on a probability space$X$was given, with the functions$f_n$satisfying some conditions which I won't specify here. The problem was to show that the$L^2 (X)$limit of the sequence is a certain given function$f(x)$. After some thought, I came up with the following "solution": I defined a sequence of functions$g_n (x,y) \in L^2 (X \times X)$and a function$g(x,y) \in L^2 (X \times X)$such that$g_n (x,x) = f_n (x,x)$and$g(x,x)=f(x)$for all$x \in X$, and I proved that$g_n (x,y) \to g(x,y) \;$in$L^2 (X \times X)$. I was sure that I havehad, in fact, proved a generalization of the proposition I was given. Embarrassingly, it took me some time to realize my mistake. I tried to salvage my proof using continuity arguments and what not, but in the end I gave up and managed to concoct a different approach, which unfortunately was a lot more complicated and messy. 1 [made Community Wiki] I was once given a question (in a take home exam) along the following lines: A sequence of complex-valued bounded measurable functions$f_n (x)$on a probability space$X$was given, with the functions$f_n$satisfying some conditions which I won't specify here. The problem was to show that the$L^2 (X)$limit of the sequence is a certain given function$f(x)$. After some thought, I came up with the following "solution": I defined a sequence of functions$g_n (x,y) \in L^2 (X \times X)$and a function$g(x,y) \in L^2 (X \times X)$such that$g_n (x,x) = f_n (x,x)$and$g(x,x)=f(x)$for all$x \in X$, and I proved that$g_n (x,y) \to g(x,y) \;$in$L^2 (X \times X)\$. I was sure that I have, in fact, proved a generalization of the proposition I was given.