Let $n \geq 2$ be an integer, and $J: \mathbf{R}^{n} \longrightarrow \mathbf{R}$ a continuous, coercive and strictly convex fonction.
For any real $\delta \geq 0$, we denote by $U_{\delta}=\mathbf{R}^{n-1} \times[\delta, \infty)$ and by $U=\mathbf{R}^{n-1} \times(0, \infty)$ . For all $k \in \mathbf{N}^{*}(k \neq 0)$, the function $J_{k}: U \longrightarrow \mathbf{R}$ defined by $$ J_{k}(\mathbf{x})=J(\mathbf{x})+\frac{1}{k \mathbf{x}_{n}}, \quad \forall \mathbf{x} \in U . $$
Prove: for $\delta>0$ small enough (to be specified), we have $$ \inf _{\mathbf{y} \in U-U_{\delta}} J_{k}(\mathbf{y})>\inf _{\mathbf{y} \in U_{\delta}} J_{k}(\mathbf{y}) $$
First, I don't understand what is the set $U-U_{\delta}$? is it $\mathbf{R}^{n-1} \times[0, \delta)$? How to continue?