I am very confused about the following:
Assume we have a sequence of functions $f_n \in$$L^1 \cap L^2 (\mathbb{R}^n)$. Then is it true that if this sequence is Cauchy both in $L^1$ and $L^2$, two limits coincide as a function in $L^1 \cap L^2$?
To make the question more general, lets say we have a space $X$ with two Hausdorff topologies. If a sequence of points which has a unique limit in both topologies, when is it guaranteed that two limits coincide?
A set $X$ can have two metrics $d$ and $d'$ (and thus Hausdorff topologies) and a sequence $(x_n)$ so that the limits $x$ and $x'$ are different. For example, take $X=[0,1]$ and let $d$ be the usual metric. Let $f:X\to X$ be the bijection that interchanges $0$ and $1$ but fixes all other points. Let $d'(a,b)=d(f(a),f(b))$. If $x_n=1/n$, then the limit points are $x=0$ and $x'=1$.
A similar argument by swapping points works for more general topological spaces as well, so there seems to be no way to guarantee a common limit without any compatibility assumptions between the two topologies.