Originally posted by JS357
Three semesters of calculus, a long time ago, here. I do remember the epsilon delta technique for establishing the limit of a function.
Intuitively the idea of uniform convergence of a sequence of functions (as opposed to a sequence of numbers, say) is simple; but at the same time it can take a long time to get comfortable with it. I searched the Internet for some nice pictures, but for my efforts only found some rather crummy ones. Anyway one explicit example of a sequence of functions that converges uniformly on the closed interval of real numbers [0,1] would be:
f_n(x) = x/n.
So the sequence is:
f_1(x) = x
f_2(x) = x/2
f_3(x) = x/3
etc.
This sequence converges uniformly to the "zero function": f(x)=0.
Let e=0.0001, say.
We could then choose N=1,000,000, say, and notice that for any n>N we have the function f_n(x)=x/n which has the following property:
For any x in [0,1] (i.e. for any real number between 0 and 1, inclusive),
f_n(x) = x/n < x/N = x/1,000,000 <= 1/1,000,000 = 0.000001 < e.
Thus, |f_n(x) - f(x)| = |f_n(x) - 0| = |f_n(x)| < 0.000001 < e, . . . . . . . (1)
We could do this for
any e>0, like even e=0.000000000000000000000000001, and
still be able to get a result like (1). We conclude that:
For any e>0 there exists some N such that, for all n>N and x in [0,1], |f_n(x) - f(x)| < e.
Therefore, by definition, the sequence {f_n} converges uniformly to f (the zero function). Some books write this as u-lim f_n = f. Uniform convergence, generally, will "preserve" properties like continuity, integrability and differentiability; so for instance if each function f_n is continuous and {f_n} converges uniformly to f, then f can be expected to be continuous. However, the idea of a "path length" (like the circumference of a circle) is itself defined by a specific limit process usually given in either the first or third semester of calculus, and the peculiar situation in the OP demonstrates that even
uniform convergence cannot be expected to preserve path length. Additional hypotheses would have to be satisfied above and beyond u-lim f_n = f to be assured that the length of the path f will equal the limit of the lengths of the paths f_n. So the OP stands as a beautiful counterexample of something in mathematics, though it is not a paradox.
Pi, after all, is simply
defined to be C/d, but little headway can be made in finding pi to some specified degree of precision until we settle on a precise definition for C itself. Euclid knew about straight-line distances, but what of distances on curves that are not straight? The ancient Greeks employed a "method of exhaustion" in which a circle was circumscribed (or inscribed) by regular polygons with an increasing number of sides. That's how they settled the matter, and the modern definition for path length is essentially the same. Interestin' stuff, for sure.
Limits of sequences aren't discussed in three semester of calculus. Usually they come up after about another year of further studying in so-called "mathematical analysis," so like I alluded to earlier the concept isn't considered "easy" as such.
EDIT: again I'm writing my post in a manner that I hope is comprehensive and may clarify things for a general audience.
Pi=4
23 Aug '11 20:48
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