= Solution
Fix $R>0$ and choose $\chi_R\in C_c^\infty((-R-1,R+1))$ with $\chi_R=1$ on $[-R,R]$. The sequence $(\chi_Rf_n)$ is bounded in $H_0^1((-R-1,R+1))$. By part (a), each subsequence has a further subsequence converging strongly in $L^2$ on this bounded interval. The original weak convergence in $L^2(\mathbb R)$ forces every such strong limit to be zero. It follows that the whole sequence satisfies
$$
\|f_n\|_{L^2([-R,R])}\longrightarrow0;
$$
otherwise a subsequence bounded away from zero would contradict the preceding compactness argument.
The pointwise hypothesis gives a uniform tail estimate:
$$
\int_{|x|>R}|f_n(x)|^2\,dx
\leq C^2\int_{|x|>R}(1+x^2)^{-2}\,dx,
$$
whose right-hand side tends to zero as $R\to\infty$, independently of $n$. Given $\varepsilon>0$, first choose $R$ so that this tail is below $\varepsilon/2$, and then choose $n$ so that the integral on $[-R,R]$ is below $\varepsilon/2$. Therefore
$$
\boxed{\|f_n\|_{L^2(\mathbb R)}\longrightarrow0.}
$$
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