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?