Let $f:\mathbb{R}\rightarrow [0,\infty]$ be a lower semi-continuous function and define the functional
$$
\begin{aligned}
F_f:&\ell^1 \rightarrow [0,\infty]\\
(x_n)_{n=0}^{\infty} &\to \sum_{n=0}^{N((x_n)_{n=0}^{\infty})} f(x_n),
\end{aligned}
$$
where $N\left((x_n)_{n=0}^{\infty}\right)=\inf\left\{
N_0\in \mathbb{N}:\, \forall n \geq N_0,\, x_{n} =0
\right\}$.  Then is $F_f$ lower semi-continuous?