Let $ p\in[1,\infty)$ and $ (f_n)_{n\in\mathbb{N}}\subset L^{p}(\mathbb{R})$ such that $ \|f_n\|_{p}\leq n^{-2}$ for all $ n\in\mathbb{N}.$ Does $ (f_n)_{n\in\mathbb{N}}$ necessarily converge pointwise a.e.? (Proof or counterexample.)

$ \textbf{Attempt:}$ I think one can construct a counterexample to this. I was considering the following construction done by Saz in this post: https://math.stackexchange.com/a/3132844/595519.

I provide the details below:

Consider the following sequence of intervals $ $ [-1,0], [0,1],\left[-2,-\tfrac{3}{2}\right], \left[-\tfrac{3}{2},-1\right], \left[-1,-\tfrac{1}{2}\right],\left[-\tfrac{1}{2},0\right],\left[0,\tfrac{1}{2}\right],\left[\tfrac{1}{2},1\right],\left[1,\tfrac{3}{2}\right],\left[\tfrac{3}{2},2\right],\left[-3,-\tfrac{8}{3}\right],\left[-\tfrac{8}{3},-\tfrac{7}{3}\right],\dots,\left[\tfrac{7}{3},\tfrac{8}{3}\right],\left[\tfrac{8}{3},3\right],\dots$ $ Denote the $ n$ th interval in the sequence by $ I_n.$ By construction, it follows that $ m(I_n)\rightarrow 0$ as $ n\rightarrow\infty.$ Now denote the characteristic function of the $ n$ th interval above by $ 1_{I_n},$ and put $ f_n(x)=n^{-3}\cdot m(I_n)^{-1/p}\cdot1_{I_n}(x)$ . It follows that $ $ \left(\int_{\mathbb{R}}|f_n(x)|^p\,dx\right)^{1/p}=n^{-3},$ $ and it follows that $ \|f_n-0\|_{p}\rightarrow 0$ as $ n\rightarrow\infty,$ so $ f_n\rightarrow 0$ in the $ L^p$ -norm.

However, I note that the sequence $ (f_n)_{n\in\mathbb{N}}$ I have constructed above fails, since it actually does converge pointwise to $ f\equiv 0,$ and even worse, it does so everywhere.

Is there any way to save this example, or should I scratch it and try something else? Finally, is there a way to use decaying exponentials in this problem to make the sequence decrease fast enough?

Thank you for time and appreciate any feedback.