TITLE: Proving that the iterated limit and the two dimensional limit are same
QUESTION [5 upvotes]: If $\lim_{(x,y)\rightarrow (a,b)} f(x,y) = L$ and if the one dimensional limits : $\lim_{x \rightarrow a}f(x,y)$ and $\lim_{y \rightarrow b}f(x,y)$ both exist, prove that :
$$\lim_{x \rightarrow a} [~\lim_{y \rightarrow b} f(x,y)] =  \lim_{y \rightarrow b} ~[~\lim_{x \rightarrow a }f(x,y)~]$$
Attempt:
$\lim_{(x,y)\rightarrow (a,b)} f(x,y) = L$ means limit exists from any direction, whether we take a curved path first along $x=a$ and then come up to $y=b$. Or, we move along $y=b$ and then come upto $x=a$. Hence, we should be able to infer that $\lim_{x \rightarrow a} [~\lim_{y \rightarrow b} f(x,y)] =  \lim_{y \rightarrow b} ~[~\lim_{x \rightarrow a }f(x,y)~]$.
However, I wanted to write down a more rigorous proof. Here's how I started off :
$\lim_{(x,y)\rightarrow (a,b)} f(x,y) = L$ means, $\forall \epsilon>0, \exists ~\delta $ neighborhood such that $|f(x,y) - l|<\epsilon$ whenever $(x,y) \in \delta$ neighborhood.
$\implies l-\epsilon<f(x,y) <l +\epsilon$ whenever $(x,y) \in \delta$ neighborhood.
Could anyone please tell me how do I proceed from here?
Thank you very much for your help in this regard.

REPLY [11 votes]: Since $\displaystyle \lim_{(x,y) \to (a,b)} f(x,y) =l$, for every $\epsilon >0$ there exists $\delta > 0$ such that if $|x-a| < \delta$ and $|y-b| < \delta$ then
$$|f(x,y) - l| < \epsilon.$$
Let 
$$l_a(y) = \lim_{x \to a} f(x,y).$$
Then with $|y-b| < \delta$,
$$|f(x,y) - l| < \epsilon \implies \lim_{x \to a} |f(x,y) - l| \leqslant \epsilon \implies |l_a(y) - l| \leqslant \epsilon.$$
Note that $\displaystyle \lim_{x \to a} |f(x,y) - l| = |l_a(y) - l|$ follows from $\displaystyle \lim_{x \to a}f(x,y)= l_a(y)$ and the reverse triangle inequality
$$||f(x,y) - l| - |l_a(y) - l|| \leqslant |f(x,y)-l_a(y)|.$$
Hence, if $|y - b| < \delta$ then $|l_a(y) - l| \leqslant \epsilon$ and we can conclude that
$$\lim_{y \to b} [\lim_{x \to a} f(x,y)] = l.$$
Now repeat the argument switching $x$ and $y$ to conclude that
$$\lim_{x \to a} [\lim_{y \to b} f(x,y)] = l.$$