Monday, September 5, 2016

Epsilon-Delta, Absolute Values, Inequalities

Working through the famed "baby" Rudin, Principles of Mathematical Analysis. (Which was not the analysis book I used in grad school: we used William Ray's Real Analysis).

First thing that dawns on me is that I'm shaky on following the formal proofs of various limits. I decide that I need to brush up on my epsilon-delta proofs, and go back to do exercises from Stein/Barcellos Calculus and Analytic Geometry. I find that I get stuck outside of simple limits of linear functions (say, with a quadratic).

Second observation is that these proofs use extensive algebra involving absolute value inequalities, and my trouble is that, in turn, I am shaky with these to the point of being blind to a lot of very basic facts. Kind of frustrating that I teach basic algebra as a career but as soon as as absolute values and inequalities enter the picture I'm nigh-helpless.

Third observation is that this explains the copious section in college algebra and precalculus texts on absolute value inequalities (which are skipped in the curriculum at our community college). Previously my intuition had failed to see the use for these; but it's precisely the skills you need in analysis to write demonstrations involving the limit definition.

A few key properties on which I had to brush up (and would argue for preparatory exercises if I was teaching/scaffolding in the direction of analysis):
  • Subadditivity: \(|a+b| \leq |a| + |b|\). Equivalent to the triangle inequality. 
  • Partial Reverse Triangle Inequality: \(|a| - |b| \leq |a - b|\). A bastardized name of my design. It follows from the preceding because \(|a| = |a - b + b| \leq |a-b| + |b|\), and then a subtraction of \(|b|\) from both sides. Gets used at the start of almost all my limit proofs. 
  • Multiplicativeness: \(|ab| = |a||b|\). Which leads to manipulations such as: multiplying both sides of an equation by a positive number allows you to just multiply into an absolute value; squaring both sides can be done into the absolute value, etc. 

More at: Wikipedia.

Discussion of general limit exercises: StackExchange.

2 comments:

  1. Why can't we just teach non-standard analysis and not bother with limit proofs?

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    Replies
    1. That thought definitely went through my mind while I was wrestling with this.

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