Double limit question

Question

Let $f: X \times Y \rightarrow \mathbb{R}$ be a continuous function where $X \subseteq \mathbb{R}$ and $Y \subseteq \mathbb{R}$. Suppose that

• $(x_{n},y_{n}) \rightarrow (x,y)$ in $\mathbb{R}^{2}$,
• $(x_{n},y_{n}) \in X \times Y$, and
• $\lim _{y_{n} \rightarrow y} f(x',y_{n}) = \infty$ for all $x' \in X$.

Is it true that there exists an $N>0$ such that $f(x_{n},y_{n}) > 0 $ for all $n \geq N$? Seems intuitive but I am a bit stuck.

Answer 1

I think it is not true. Suppose f(x,y)=-x+ 1/|y-1|. let y=1, yn=1+1/n, xn=2n.

Answer 2

No, this is not necessarily true. The function $f(x,y)=\frac{1}{y}-\frac{1}{x^2}$ is continuous in $(0,1)^2$, and for $x_n=y_n=\frac{1}{n}$, we have $f(x_n,y_n)<0$ for $n\geq 2$, while for $x'\in(0,1)$, we have $f(x',y_n)=n+1-\frac{1}{x'^2}\to\infty$.

Edit: To have a counterexample with $f$ bounded below, for $(x,y)\in (0,1)^2$, consider the function $$f(x,y)=\left\{\begin{array}{c l}\frac{1}{y}-\frac{1}{x}-1, & 0<y<x<1 \\ -1, &\text{otherwise}, \end{array}\right.$$ and choose $x_n=y_n=\frac{1}{n}$.