Support of the limit of a convergent sequence of distributions

Question

I read that if $u_n \to u \in \mathcal{D}'(X)$, then $$ \text{supp} \, u \subset \bigcap_{n \geq 1} \bigcup_{m \geq n} \text{supp} \, u_m. $$ However, the proof given shows that $$ \text{supp} \, u \subset \bigcap_{n \geq 1} \overline{\bigcup_{m \geq n} \text{supp} \, u_m}. $$ Since it seems to me that the set in the second inclusion is larger than the one in the first inclusion, I would like to know if there is a mistake or if there is a reason why the second inclusion would imply the first one ?

Answer 1

It seems that the first inclusion (without a closure) is false.

Try the following counterexample in $\mathcal{D}'(\mathbb{R})$. Consider the sequence $u_n=\delta_{1/n}$, there $\delta_a$ denotes the Dirac function defined by $\delta_a(f):=f(a)$.