A Kick in the Discovery

Let’s get straight into this one today!

The Triangle Inequality

As the name suggests, this is an equality that can be related back to a triangle. Consider the triangle’s sides below,

The triangle inequality states: the sum of any two side lengths of a triangle is greater than or equal to the third side. In the triangle above the sum of the lengths and will have to be larger than or equal to the third side

For triangles, this is intuitive. If you had to walk from the top of the triangle to the bottom right, the shortest path would be to travel along . If we used the language (and notation) of vectors, we would write this as: The same holds true for real numbers too! This captures in the triangle inequality:

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Theorem: (The Triangle Inequality) Let Then,

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Comment: This probably seems obvious, I mean However, the real fun happens when we have negative numbers e.g.

Scratch Work: How might we go about this? We will need to use some properties of inequalities. For a refresher,

We’ll we can see that:

Lemma 1:

Proof: We can prove this case by case. When then Likewise, when we have

Lemma 2: If for then

Proof: Let We do a proof by cases, yet again.

Case 1: Here, we automatically have from the right-hand side of the inequality.

Case 2: This case requires a little more work. When we have This implies,

Looking at the left side of the inequality we see,

In either case we have

We have enough to prove the triangle inequality! See what you can come up with before you continue!

Proof of the Triangle Inequality:

Let By lemma 1 we know both,

and

Adding these we deduce,

Which, by lemma 2 implies the triangle inequality:

Closing Remarks

I don’t think it would be a hyperbole if we said that the triangle inequality is the most important inequality for real analysis. Nuf said.

This article was purposely short, so that way the people who only want to learn about the triangle inequality won’t get bogged down with more than they needed! But, for those reading the series in part 2 of this week’s posts we use the triangle inequality to show that when a sequence has a limit, it must be unique! Check it out here!