Is this a valid move in my simple inequality proof?

Question

Prove: $|x+y+z| \leq |x|+|y|+|z|/$

We have the theorem 1: $|a+b| \leq |a|+|b|$.

Let $a = x$ and $b = (y+z)$; thus by theorem 1 it follows: $|x+y+z| \leq |x|+|y+z|$,

$$|x+y+z|-|x| \leq |y+z|.$$

We know by theorem 1: $|y+z| \leq |y|+|z|$.

Thus we have: $|x+y+z|-|x| \leq |y+z| \leq |y|+|z|$.

Therefore by way of transitivity, we have: $|x+y+z|-|x| \leq |y|+|z|$.

Thus, $|x+y+z| \leq |x|+|y|+|z|$.

What is this « equal » sign, here? Why do you think you cannot judge the validity of your proof by yourself? — Plop, Dec 02 '22 at 22:11
I just started learning proof writing, so I naturally have doubts judging validity. Which sign are you referring to? — lopan, Dec 02 '22 at 22:14
I removed the stray equals sign. Question for you to consider: was it really necessary to move $|x|$ to the left-hand side? — 311411, Dec 02 '22 at 22:23
Definitely not necessary, but I wanted to make it explicit that I was going to invoke theorem 1 a second time. Without moving the $|x|$ out of the way, I thought it would be unclear how that side plugs back into theorem 1. — lopan, Dec 02 '22 at 22:27

score 1 · Answer 1 · answered Dec 02 '22 at 22:34

1

You can apply the triangle inequality twice to get the desired result:

\begin{align*} |x + y + z| & = |(x + y) + z|\\\\ & \leq |x + y| + |z|\\\\ & \leq |x| + |y| + |z| \end{align*}

and we are done.

Hopefully this helps!

answered Dec 02 '22 at 22:34

Átila Correia

I think I see now, the second application of triangle eq is really inline, or "beside" the $|z|$. I was (maybe still am) confused here since if the triangle eq was only stating equality, then I see how essentially substituting $|x+y|$ for $|x|+|y|$ inline makes sense, but since triangle eq is an inequality, I'm not clear on how the inline substitution is still good -- I mean, I get that it is, but I don't know how to think about it. – lopan Dec 02 '22 at 22:47
1

Ah, I see now. Could think of it simply as adding $|z|$ to both sides of triangle eq which is obviously good. – lopan Dec 02 '22 at 22:52

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