A property of $\Gamma$-convergence

Question

In the wikipedia of $\Gamma$-convergence, there is a property "A constant sequence of functionals $F_n=F$ does not necessarily $\Gamma$-converge to $F$, but to the relaxation of $F$, the largest lower semicontinuous functional below $F$". I am confused about this property. It is clear to me that the $\Gamma$-limit of $F_n$ should be the largest lower semicontinuous functional below $F$ by the definition of $\Gamma$-convergence and the fact that the $\Gamma$-limit is lower semicontinuous. However, why in this case the largest lower semicontinuous functional below $F$ is not $F$ itself? If we consider the simplest situation $F_n$ and $F$ defined on $\mathbb{R}$, then as $F$ is a constant function we must have $F$ is continuous, and so it is lower semicontinuous. Thus as a lower semicontinuous function, its relaxation should be itself. Do I have misunderstanding?