Ekeland's Variational Principle for Upper Semicontinuous Functions on a Complete Metric Space

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space that is complete, and let $F:X\to\mathbb{R}$ be upper semicontinuous on $X$ and bounded above. Let $\eta$ and $\kappa$ be positive reals, and let $x_{0}\in X$ satisfy

Then there is $\bar{x}\in X$ with the following three properties.

1. (Value)¶ $F(\bar{x})\ge F(x_{0})$.

2. (Distance)¶ $d(\bar{x},x_{0})\le\kappa$.

3. (Strict perturbed maximum)¶ $F(x)-\frac{\eta}{\kappa}\,d(x,\bar{x})<F(\bar{x})$ for every $x\in X$ with $x\ne\bar{x}$.