Let $\ell^1(\mathbb{R}^d)$ be the Banach space of sequences in $\mathbb{R}^d$ for which the norm $\|(x_n)_n\|=\sum_{n=1}^{\infty} \|x_n\|_2$ is finite. Let $f:\mathbb{R}^n\rightarrow [0,\infty)$ be non-constant, lower semi-continuous, and coercive. So it admits a minimizer by Tonelli's direct method.
Under what additional (growth?) conditions on $f$ can I guarantee that there is a sequence (of sequences) $\{(x_n^N)_n\}_N$ in $\ell^1(\mathbb{R}^n)$, such that $x_n^N=0$ for every $n>N$ and the the coercivity-type condition is satisfied: $$ \sum_{n=1}^N f(x_n^N) = \inf_{(y_n)_n\in \ell^1(\mathbb{R}^n)} \sum_{n=1}^{\infty}f(y_n) + \mathscr{o}(1) \mbox{ as } N\to \infty. $$
Update: I managed to show that it is enough for the map $x_n\mapsto \sum_{n=1}^{\infty}(x_n)$ to have compact sub-level sets...and by Grothendieck's result this means that the set $$ \left\{ (y_n)_n \in \ell^1(\mathbb{R}^d):\, \sum_{n=1}^{\infty} f(y_n)\leq s \right\}\subseteq \overline{co\left( \{(z_n)_n\}_{N \in \mathbb{N}} \right)}, $$ where $\|(z_n^N)\|_{\ell^1(\mathbb{R}^d)}\to 0$ as $N\to \infty$. However, I cannot manage to relate this condition to $f$..