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The question I have may not be completely well-defined. I'm in the process of proving that the square metric on $\mathbb{R}^n$ is in fact a metric, the final statement of which is $$ \max\limits |p_i - q_i| \leq \max\limits |p_i - r_i| + \max |r_i - q_i| \qquad (\star). $$ The first line is $$ |p_i - q_i| \leq |p_i - r_i| + |r_i - q_i|. $$ In the next line, I want to say something to the effect of "taking the maximum over $i$, because the max function is increasing, we get ($\star$)." I'm not completely sure that I can do this, because the maximum isn't necessarily a "function" (or is it?). I can treat it, in some sense, as a function of the inputs $|p_i - q_i|$, for example, or even as a function of $i$, though the latter wouldn't necessarily be increasing (e.g., $|p_1 - q_i|$ could be maximal.)

Is there a different way to phrase this justification, or is it just wrong?

Mathematical Endeavors
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I think you are stumbling toward the right idea, but what you say is somewhere between incomprehensible and wrong.

Hint. What can you say if each of the three maxima occurs at the same index $i$? How does that help you when the indices differ?

Ethan Bolker
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  • I've thought about this, and I'm really not sure. If the three maxima were to occur at the same $i$, the statement would immediately follow since I showed the inequality $|p_i - q_i| \leq |p_i - r_i| + |r_i - q_i|$ held for every $i$. If the indices were to differ, I can't substitute directly into the inequality unless I were to bound the right hand side first. – Mathematical Endeavors Jan 23 '23 at 14:43
  • Think about an inequality you can write for the right side for the index $i$ that maximizes the left side. – Ethan Bolker Jan 23 '23 at 14:51
  • I can say $|p_i - r_i| \leq \max |p_i - r_i|$ and $|r_i - q_i| \leq \max |r_i - q_i|$, so $|p_i - r_i| + |r_i - q_i| \leq \max |p_i - r_i| + \max |r_i - q_i|$. This makes sense to me and is how I ended up solving the problem, though I'm not sure how to relate this to "maximizing over $i$." – Mathematical Endeavors Jan 23 '23 at 17:04
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    That is how to do it. So for you, "maximizing over $i$" was a hint that confused more than it helped. – Ethan Bolker Jan 23 '23 at 19:22