Consequences of $f_n \rightharpoonup 0 $ in $W^{1,2}(\mathbb{R}^2)$

Question

I am trying to establish is the following are correct or not.

Let $f_n \rightharpoonup 0 $ in $W^{1,2}(\mathbb{R}^2)$. Then:

1. $f_n$ are periodic functions

2. $f_n \rightarrow 0$ uniformly

3. $\frac{\partial f_n }{\partial x}+\frac{\partial f_n }{\partial y} \rightharpoonup 0$ in $L^2(\mathbb{R}^2)$

4. $f_n \rightarrow 0 \Rightarrow \lVert \nabla f_n \rVert_{L^2} \rightarrow 0 $

My attempt is:

1. False. Periodic functions $\psi$ do not satisfy $\lVert \psi \rVert_{L^2} < \infty$

2. I don't know. For Rellich-Kondrakov theorem $f_n$ converges in $L^q$ with $1 \le q < p^*$, but how can I say that the sequence converges in $C(\Omega)$?

3. True. $f_n \rightharpoonup 0 $ in $W^{1,2}(\mathbb{R}^2) \iff D^{\alpha}f_n \rightharpoonup 0 $ in $L^2(\mathbb{R}^2) \quad \forall |\alpha| \le 1$

4. I don't know, I think it's false because the $L^2$ gradient is not related to uniform convergence, but I cannot find a counterexample.

Can anyone help me? Thank you in advance.