How would I prove that if $f: \mathbb R^2 \to \mathbb R$ is a function such that
$$\lim_{(x,y)\to(a,b)} f(x,y) = L$$
and for every $y_0 \in \mathbb R$
$$ \lim_{x\to a} f(x,y_0) = L'_{y_0}$$
and for every $x_0 \in \mathbb R$
$$ \lim_{y\to b} f(x_0,y) = L''_{x_0}$$
then
$$ \lim_{x\to a}\left(\lim_{y\to b} f(x,y)\right) = \lim_{y\to b}\left(\lim_{x\to a} f(x,y)\right) = L$$