Showing the multi-variable limit exists using the epsilon-delta definition of a limit

Question

How does one go about the following proof?

Let $f : \mathbb{R}^2\rightarrow \mathbb{R}^3$ be given by $(x,y) \mapsto (x, x+y, x-y)$. Show that $$\lim_{(x,y)\rightarrow (1,0)}f(x,y)=(1, 1, 1).$$ I'm having trouble with the proof because of the different dimensions although I know we use $\left \| \right \|$ instead of $\left | \right |$ for $\mathbb{R}^n$. Can someone please construct a formal proof for this?

Answer 1

If $|x-1|<\delta$ and $|y|<\delta$, then

\begin{align} \|f(x,y)-(1,1,1)\|_2^2 &= (x-1)^2 + (x+y-1)^2 + (x-y-1)^2\\ &\le (x-1)^2 + (|x-1|+|y|)^2 + (|x-1|+|y|)^2\\ &\le 9\delta^2. \end{align}