Inequality connecting inf and liminf

Question

Suppose $f(x,y)$ is a continuous, nonnegative-valued and bounded function on $\mathbb{R}^2$. Is the following correct? $$ \inf_{x,y} f(x,y)\le \liminf_{y\to\infty}\inf_{x}f(x,y) $$

Answer 1

$$\inf_{x,y} f(x,y)\leq \inf_{x}f(x,y)$$ and thus $$\inf_{x,y}f(x,y)=\liminf_{y\to\infty }\inf_{x,y}f(x,y)\leq \liminf_{y\to\infty }\inf_x f(x,y)$$

what conclude the proof.