Lebesgue integral $\geq1$ for $f_n \rightarrow 0$ uniformly in every compact set of $ \mathbb{R}$

Question

Is this true or false?

$\exists f_n:\mathbb{R} \rightarrow \mathbb{R}^+, n\in \mathbb{N}$ sequence of Lebesgue integrable functions such that $f_n \rightarrow 0$ uniformly in every compact $K \subseteq \mathbb{R} $ but $\int_\mathbb{R} f_n dλ \geq1 $ for every $n\in \mathbb{N}$.

Answer 1

I think for $f_n(x)=\cases{1/n, &if $x \in [0,n]$ \cr 0 & otherwise \cr} $ I have what I want.