A couple weeks ago I wrote about the Weierstrass approximation theorem, the theorem that says every continuous function on a closed finite interval can be approximated as closely as you like by a polynomial.

The post mentioned above uses a proof by Bernstein. And in that post I used the absolute value function as an example. Not only is |*x*| an example, you could go the other way around and use it as a step in the proof. That is, there is a proof of the Weierstrass approximation theorem that starts by proving the special case of |*x*| then use that result to build a proof for the general case.

There have been many proofs of Weierstrass’ theorem, and recently I ran across a proof due to Lebesgue. Here I’ll show how Lebesgue constructed a sequence of polynomials approximating |*x*|. It’s like pulling a rabbit out of a hat.

The staring point is the binomial theorem. If *x* and *y* are real numbers with |*x*| > |*y*| and *r* is any real number, then

.

Now apply the theorem substituting 1 for *x* and *x*² – 1 for *y* above and you get

The partial sums of the right hand side are a sequence of polynomials converging to |*x*| on the interval [-1, 1].

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If you’re puzzled by the binomial coefficient with a top number that isn’t a positive integer, see the general definition of binomial coefficients. The top number can even be complex, and indeed the binomial theorem holds for complex *r*.

You might also be puzzled by the binomial theorem being an infinite sum. Surely if *r* is a positive integer we should get the more familiar binomial theorem which is a *finite* sum. And indeed we do. The general definition of binomial coefficients insures that if *r* is a positive integer, all the binomial coefficients with *k* > *r* are zero.