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.