Let $f_n=\sqrt{n}e^{-nx}$ on $(0,1)$ does $\int_{0}^{1}f_nh \to 0$ for all $h \in L^1$

Question

Let $f_n=\sqrt{n}e^{-nx}$ on $(0,1)$ does $\int_{0}^{1}f_nh \to 0$ for all $h \in L^1$

This is the last part of a problem I am solving. These are the previous parts i have shown:

1) $f_n \to 0$ but not uniformly

2) $\|f_n\|_2 \to 0$

3) $f_n \rightharpoonup 0 $ In $L ^2$

I am pretty sure the statement is false but I do not know how to prove it. It is a weaker statement than weak convergence in $L^{\infty}$ so I don't have any standard theorems to apply.