real analysis - Show $lim_{m to infty ,n to infty } f(frac{{leftlfloor {mx} rightrfloor }}{m},frac{{leftlfloor {ny} rightrfloor }}{n}) = f(x,y)$

Suppose $f(x,y)$ is defined on $[0,1]\times[0,1]$ and continuous on each dimension, i.e. $f(x,y_0)$ is continuous with respect to $x$ when fixing $y=y_0\in [0,1]$ and $f(x_0,y)$ is continuous with respect to $y$ when fixing $x=x_0\in [0,1]$. Show

$$\lim_{m \to \infty ,n \to \infty } f\left(\frac{{\left\lfloor {mx} \right\rfloor }}{m},\frac{{\left\lfloor {ny} \right\rfloor }}{n}\right) = f(x,y)$$

My attempt:

First, I know $$\lim\limits_{m \to \infty ,n \to \infty } \left(\frac{{\left\lfloor {mx} \right\rfloor }}{m},\frac{{\left\lfloor {ny} \right\rfloor }}{n}\right) = (x,y)$$

Secondly it looks

$$\lim\limits_{m \to \infty }\lim\limits_{n \to \infty } f\left(\frac{{\left\lfloor {mx} \right\rfloor }}{m},\frac{{\left\lfloor {ny} \right\rfloor }}{n}\right) = \lim \limits_{m \to \infty } f\left(\frac{{\left\lfloor {mx} \right\rfloor }}{m},y\right) = f(x,y)$$

and
$$\lim\limits_{n \to \infty } \lim\limits_{m \to \infty } f\left(\frac{{\left\lfloor {mx} \right\rfloor }}{m},\frac{{\left\lfloor {ny} \right\rfloor }}{n}\right) = \lim\limits_{n \to \infty } f\left(x,\frac{{\left\lfloor {ny} \right\rfloor }}{n}\right) = f(x,y)$$

since $f(x,y)$ is continuous on each dimension.

However, I am not sure if this can infer $\lim\limits_{m \to \infty ,n \to \infty } f(\frac{{\left\lfloor {mx} \right\rfloor }}{m},\frac{{\left\lfloor {ny} \right\rfloor }}{n}) = f(x,y)$.

Can anyone provide some help? Thank you!

Added:

I am now sure $\lim\limits_{m \to \infty } \lim\limits_{n \to \infty } {a_{mn}} = \lim\limits_{n \to \infty } \lim\limits_{m \to \infty } {a_{mn}} = L$ does not imply $\lim\limits_{m \to \infty ,n \to \infty } {a_{mn}} =L$ in general. Hope someone can help solve the problem.