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I always understood the triangle inequality to be the following:

If you have 2 sides of a triangle, say 4 and 6, the third side had to be between 2 and 10.

However, I see the triangle inequality written in this form:

$|a+b| \leq |a|+ |b| $

How do you prove it?

I know that:

$-|a| \leq a \leq |a|$

and

$-|b| \leq b \leq |b|$

and adding you get:

$-(|a| + |b|) \leq a + b \leq |a| + |b|$

Now what?

I am learning this because I want to learn the Sum Law proof:

$\lim_{x \to a} [f(x) + g(x)] = L + M$

The first step is showing that

$|f(x) + g(x) - (L + M)| \leq |f(x) - L| + |g(x) - M|$ using epsilon delta

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    $\begingroup$ Hint: $|x| \le M$ if and only if $-M \le x \le M$. $\endgroup$ Jan 5 '18 at 23:11
  • $\begingroup$ The third side is between $2$ and $10$ means that $|b|-|a| \le |a+b| \le |a| + |b|$ for $a=4$ and $b=6$. $\endgroup$
    – Math Lover
    Jan 5 '18 at 23:14
  • $\begingroup$ but |b|-|a| does not equal -(|a| + |b|) $\endgroup$
    – Jwan622
    Jan 6 '18 at 21:14
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$$-(|a| + |b|) \leq a + b \leq |a| + |b|$$ implies $$|a+b|\le |a|+|b|$$

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  • $\begingroup$ why is this? are you just ignoring the first part of the inequality? $\endgroup$
    – Jwan622
    Jan 6 '18 at 19:34
  • $\begingroup$ If a number is between -b and b for a positive b, then absolute value of the number is less than or equal b. I am not ignoring the left side at all. $\endgroup$ Jan 6 '18 at 19:46
  • $\begingroup$ ohhh. <More letters> $\endgroup$
    – Jwan622
    Jan 6 '18 at 21:11
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Consider that $$ |x+y| \leq |x|+|y| \Leftrightarrow |x+y|^2 \leq (|x|+|y|)^2 $$ $$\Leftrightarrow x^2+2xy+y^2 \leq x^2+2|x|.|y|+y^2 $$ $$\Leftrightarrow xy \leq |x|.|y| $$ $$\Leftrightarrow xy \leq |xy| $$

The last inequality is always true.

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Take $|a+b|\leq |a|+|b|$ and replace "$a$" with "$x-y$" and "$b$" with "$y-z$":

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

Interpret something like $|x-y|$ as the distance between $x$ and $y$, where $x,y$ do not have to be real numbers, and $|\cdot|$ does not have to be the standard Euclidean metric.

This is the general idea of the triangle inequality, and it now applies to your standard example of side lengths. Interpret $x,y,z$ as the points of a triangle on the plane, and so e.g. $|x-y|$ is the length of the side between the points $x$ and $y$. The statement now says, "the combined length of two sides of a triangle must be longer than the 3rd side".

I also prefer to share this form of the inequality when presenting it to students, as well, because there is a useful, colloquial heuristic for remembering it: Stopping by Starbucks on your way to class cannot be a shortcut.

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