Basically I'm trying to prove a limit with the delta epsilon definition and I'm reading about how to do it. In one example, the book does the following:
$\lim_{(x,y)\to (1,-1)} \frac{y+1}{(3(x-1)^2+2(y+1)^2)^{1/3}} \leq \lim_{(x,y)\to (1,-1)} \frac{\lVert[x-1,y+1]\rVert}{(2\lVert[x-1,y+1]\rVert ^2+2\lVert[x-1,y+1]\rVert^2)^{1/3}}$
I don't understand why this is true. I know for a fact that $(y+1) \leq ||[x-1,y+1]||$ but I don't understand what's going on in the denominator. As far as I know the denominator of the first function should be greater than or equal to the one on the second function for this inequality to hold true, but I don't understand why it works here.
Edit: $\lVert [x-1,y+1] \rVert^2$ is equal to $\sqrt {(x-1)^2+(y+1)^2} $