Let $\{f_k\}_k\in L^{1}_{loc}(\mathbb R)$ be a sequence of real value functions such that $\mathrm{supp}f_k\subset \{|x|\leq\frac{1}{k}\}$ and $\int f_k(x) dx = 1 \,\, \forall k$.
How to prove that $\{f_k^2\}_k$ doesn't converge in distribution? That is, for some $\phi\in C^{\infty}_c(\mathbb R)$, $\lim_{k\rightarrow \infty} \int f_k^2(x)\phi(x) \,dx $ doesn't exist.
Intuitively I tried letting $F_k(x)= f_k(1/k)/k$ just like when dealing with dirac funtion, then get $\int F_k(x)\, dx =1$ and $\int f^2_k(x)\, dx= k\int F_k^2(x)\, dx$, but it doesn't seem working.