Understanding Triangle Inequality and applying it

[MJC684]

I was wondering if anyone knew of a text or website that has a very indepth explanation of the triangle inequality and its variations really just how to apply it with limit proofs. I mean i understand it in terms of a triangle as no one side can be longer than the sum of the two other sides, but I am more interested in the explanation in terms of the real line. Also what about this max(a,b) and min(a,b) notation i keep seeing? I never seen it used in my precalculus studies. I am guessing that it denotes the largest or smallest member of a set? Is that really all there is too it?

Basically i want to master all the prerequiste skills that i will need to seriously study and understand limits. Any advice would be great.

 

[pbandjay]

Yes the min{.} and max{.} just refer to the minimum and maximum element of the set. Sometimes, the min or max does not exist, though.

How exactly are you trying to apply the triangle inequality where you are confused? In some proofs, it might just require substitution, being careful of the direction of the inequality.

The triangle inequality is a feature that needs to be verified for any metric space. If (X,ρ) is a metric space, then, for z,w,ξ ∈ X, one has ρ(z,w) ≤ ρ(z,ξ) + ρ(ξ,w), which attains equality if ξ lies on the segment [z,w]. Depending on the metric space, this segment isn't necessary on a straight Euclidean line.

In the case of the real line, the metric space is (R,|.|) and for a,b ∈ R, one has |a+b| ≤ |a| + |b|.

http://en.wikipedia.org/wiki/Triangle_inequality
http://en.wikipedia.org/wiki/Metric_space

 

[snipez90]

Well yes, that is what the max and min stand for, but there is definitely more to it in analysis. Both the max/min notation and the triangle inequality allow you to make estimates. If you have the inequality |x-a| < min(epsilon, 1), then the most important thing this inequality is telling you is that |x-a| < epsilon AND |x-a| < 1 (Why?). The most basic application of this used to prove limits of quadratic equations, where sometimes it is convenient to make the simplifying assumption that $\delta \leq 1$ so that |x-a| < 1 (if x is approaching a). Then this assumption will typically allow you to find a bound on |x-a| in terms of epsilon. Then to justify the use of the assumption that $\delta \leq 1$, we use the min notation |x-a| < min((some positive factor)*epsilon, 1). Similarly, the max notation is often useful when we're dealing with limits approaching infinity.

The triangle inequality really depends on the fact that you can add and subtract the same number to some other quantity for free since it's the same as adding 0. Ideally, you'll already have made some decent estimates in your problem, and you just need to tie them together (this is completely analogous to the situation where you needed max/min). For instance, if you found that |f(x) - L| < epsilon and |g(x) - L| < epsilon, and you wanted to find an estimate on |f(x) - g(x)|, then note that you can introduce L in the last expression by noting that

$$|f(x) - g(x)| = |f(x) -L + L - g(x)| \leq |f(x) -L| + |L - g(x)| = |f(x) -L| + |g(x) - L| < \varepsilon + \varepsilon.$$

Application of the triangle inequality is very intuitive, since all we're saying is that if f is close to L, and g is close to L, then f and g should be near each other as well. For further reading, I suggest trying to find a copy of Spivak's Calculus text and reading chapter 5 (google books might have it), or some other intro analysis text.