# Multivariable $\epsilon-\delta$ proof verification

Prove that $\lim\limits_{(x,y) \to (1,1)} xy=1$

Of course, I am aware that this is "obvious", but I want to add some rigor to it. When I searched around for multivariable limits using $\epsilon-\delta$, most of the examples had $(x,y) \rightarrow (0,0)$, but in this case I have $x$ and $y$ approaching something else.

$(x,y) \rightarrow (1,1) \Leftrightarrow \lvert\lvert (x,y)-(1,1)\lvert\lvert \rightarrow 0$ which can be written as $0 < \sqrt {(x-1)^2+(y-1)^2} < \delta$ for some arbitrarily small $\delta >0$.

Goal: show that $\forall$ $\epsilon>0$ $\exists$ $\delta>0$ such that

$0 < \sqrt {(x-1)^2+(y-1)^2}<\delta\Rightarrow0<|xy-1|<\epsilon$

Proof:

If $0 < \sqrt {(x-1)^2+(y-1)^2}<\delta$, then

$|xy-1|=|xy-x-y+1+x+y-2|=|(x-1)(y-1)+(x-1)+(y-1)|$

$\le|(x-1)(y-1)|+|x-1| +|y-1|=|x-1||y-1|+|x-1|+|y-1|$

$=(\sqrt{(x-1)^2})(\sqrt{(y-1)^2})+\sqrt{(x-1)^2}+\sqrt{(y-1)^2}$

$\le(\sqrt{(x-1)^2+(y-1)^2})^2+2\sqrt{(x-1)^2+(y-1)^2}<\delta^2+2\delta$