SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} F(f)\geq \limsup_{M}F_{M}(f_{M}) \\ \geq \limsup_{M_{\ell}}F_{M_{\ell}}(f_{M_{\ell}}) \\ \geq \liminf_ {M _ {\ell}} F _ {M _ {\ell}} \left(f _ {M _ {\ell}}\right) \\ \geq \liminf_{M_{\ell}}F_{M_{\ell}}(f_{M_{\ell}}^{*}) \\ \geq F (f ^ {*}), \\ \end{array}
$$
where we used the lim sup-inequality of  $\Gamma$ -convergence in the first step, standard properties of lim sup and lim inf in the second and third step, the fact that  $f_{M_\ell}^*$  is a minimizer of  $F_{M_\ell}$  in the fourth step, and finally the lim inf-inequality of  $\Gamma$ -convergence. Since  $f \in H_k$  was arbitrary, this shows that  $f^*$  is a minimizer of  $F$ .
Furthermore, let  $(f_M)_M$  be a recovery sequence for  $f^{*}$ , then
$$
\begin{array}{l} F(f^{*})\geq \limsup_{M}F_{M}(f_{M}) \\ \geq \limsup_{\ell}F_{M_{\ell}}(f_{M_{\ell}}) \\ \geq \limsup_{\ell}F_{M_{\ell}}(f_{M_{\ell}}^{*}), \\ \end{array}
$$
where we used the lim sup-inequality in the first step, an elementary property of lim sup in the second step, and finally that  $f_{M_{\ell}}^{*}$  is a minimizer of  $F_{M_{\ell}}$ . Since  $f_{M_{\ell}}^{*} \xrightarrow{\mathcal{P}_{1}} f^{*}$ , the lim inf-inequality of  $\Gamma$ -convergence implies that
$$
F (f ^ {*}) \leq \operatorname * {l i m i n f} _ {\ell} F _ {M _ {\ell}} (f _ {M _ {\ell}} ^ {*}),
$$
so we find that
$$
\lim  _ {\ell} \sup  _ {F _ {M _ {\ell}} (f _ {M _ {\ell}} ^ {*})} F _ {M _ {\ell}} (f _ {M _ {\ell}} ^ {*}) \leq F (f ^ {*}) \leq \lim  _ {\ell} \inf  _ {F _ {M _ {\ell}} (f _ {M _ {\ell}} ^ {*})},
$$
establishing that  $F_{M_{\ell}}(f_{M_{\ell}}^{*}) \to F(f^{*})$