When are the sup and euclidean metric interchangeable in analysis on $\mathbb{R}^{n}$ In Analysis on Manifolds, Munkres makes the statement that for most purposes, the sup metric, which is $$max\{|x_{i} - y_{i}| \space i \in \{1 \dots n\}\}$$ and the euclidean metric are "equivalent." In other words, they can be interchanged in theorems like this:

I cannot find an instance when interchanging the metrics in a theorem is unacceptable -- I suspect that this has to do with the fact that one can place, within any neighborhood of one metric in $\mathbb{R^{n}}$, another neighborhood of the second metric.
Is my hunch correct -- that since neighborhoods formed using either metric can be placed within each other, all theorems involving a metric can be formulated using either metric? At least with respect to analysis on $\mathbb{R^{n}}$?

The Euclidean norm is strictly convex, and the sup norm isn't.  This is relevant when unique best approximations are relevant.  For example, given a closed convex subset $X$ of $\mathbb R^n$, and a point $p\in\mathbb R^n$, there is a unique $x\in X$ with smallest Euclidean distance to $p$.  This is not true with the sup norm; e.g., in $\mathbb R^2$, if $X$ is the closed unit "ball" in the sup norm, and $p=(2,0)$, then the smallest distance from $p$ to $X$ is $1$, but this minimum is achieved for all of the infinitely many points $(1,t)\in X$.  
