I am teaching myself multivariable real analysis from Zorich's Math Analysis II.
I am trying to prove that $$f(x)=\begin{cases} x+y\sin(1/x), \text{if } x\neq0\\ 0, \text{if } x=0 \end{cases}$$ satisfies $\lim_{(x,y)\to(0,0)} \; f(x,y)=0$.
My proof: Let $\epsilon>0$ and consider the ball of radius epsilon $B_{\epsilon}(0)$. Choose $\delta>0$ such that $\delta<\epsilon/2$. By norm equivalence, we may consider $0<\|(x,y)\|_{\infty}<\delta$. For all such $(x,y)$, we have $$\|f(x,y)-0\|=\|x+y\sin(1/x)\|\leq\|x\|+\|y\|\|\sin(1/x)\| \leq 2\|(x,y)\|_{\infty} < 2\delta <\epsilon)$$
since $\|\sin(1/x)\|\leq 1$ for all $x$.
My only problem is justifying why I can use the sup-norm or max-norm instead of the Euclidean or $p=2$ norm. Please verify my proof if possible, as self-study is difficult.
There is a result that you might know that states that for finite dimensional spaces, all norms are equivalent.
If you don’t know it, you’ll be able to prove that $$\sup(\vert x \vert , \vert y \vert)\le \sqrt{x^2+y^2} \le \vert x \vert + \vert y\vert \le 2 \sup(\vert x \vert , \vert y \vert)$$ for all $x,y \in \mathbb R$