Fundamental Theorem of Gamma-Convergence

Question

Let $(X, d)$ be a metric space and $F_\varepsilon, F\colon X\to [-\infty, \infty]$. Suppose $F_\varepsilon$ is an equicoercive sequence of functions on $X$, i.e. for all $t\in\mathbb{R}$ there exists a compact set $K_t$ such that $\{ x\colon F_\varepsilon(x)\le t \} \subset K_t$ for every $\varepsilon > 0$. Suppose $F_\varepsilon\xrightarrow{\Gamma} F$, i.e. for all $u\in X$ we have that

1. for every sequence $\{u_\varepsilon\}$ converging to $u$ it holds that $F(u)\le\displaystyle\liminf_{\varepsilon\to 0}F_\varepsilon(u_\varepsilon)$.
2. there exists a sequence $\{u_\varepsilon\}$ converging to $u$ such that $F(u) = \displaystyle\lim_{\varepsilon\to 0}F_\varepsilon(u_\varepsilon)$.

Question: Is $F$ coercive on X?

Answer 1

Yes. Take any sequence $\{x_i\}$ for which $F(x_i)$ is bounded. For each $x_i$ consider the sequence satisfying (2), $\{x^\epsilon_i\}$ s.t. $x^\epsilon_i\to x_i$. Consider $x_i^{\epsilon_i}$ such that $F_{\epsilon_i}(x_i^{\epsilon_i})$ is bounded. Due to equicoercivity, $x_i^{\epsilon_i}$ has to be in a compact set, so it has a converging subsequence, thus $x_i$ must also have a converging subsequence.

Comment: equicoercivity is essential, e.g., take $F_\epsilon=\epsilon x^2$. $F_\epsilon\stackrel{\Gamma}{\to}0$, which is not coercive.