[dmurray]

> The union V_n = U_1 + U_2 + ... + U_n has combined length 1 - 1/2*n < 1, so it can't contain [0, 1].

This argument seems way less convincing to me than the diagonalization argument, because as n is going to infinity that length does become 1.

[clintonc]

Then use 1/3 instead of 1/2 for a combined length of 2/3 -- the total length of the intervals can be as small as you like. This hints at the fact that any countable subset of the real numbers is Lebesgue measure zero.

Even using 1/2, the set that remains is nonempty due to the Cantor intersection theorem. The total length of the intervals is 1, which means that the remainder has no "interior" (i.e., contains no open interval), but the converse is not true: removing intervals whose lengths sum to less than one does not mean that the remainder will contain any interval. This is the consideration that allows you to create what are called "fat Cantor sets" -- the middle thirds Cantor set has Lebesgue measure zero, but by removing smaller intervals you can get other, homeomorphic sets that have positive measure.

[gowld]

Then let U_i be the interval of length 1/3^i centered on f(i), so that the total length is 1/2, far less than 1.

Even though the supposed "surjection" is infinite, it's still the case that every x in [0,1] would be in in one of the finite U_n and therefore V_n. But every K_n clearly has measure > 0 and is therefore non-empty, and since the K_n are nested subsets, there is at least one special point x_omega that is in all of the K_n.

The "intuitive" problem (not logical problem) with PP's proof is that it relies on measure and completeness, which is far more technologically complex than the decimal diagonizalization argument.

Here is intuitive "rebuttal": the same proof strategy seemingly proves that the rationals are uncountable! (This is of course technically false, because rational intervals are incomplete and all have measure 0 in the first place. But understanding this is much more complicated than imagining an 2-D infinite spreadsheet of decimal numbers between 0 and 1.)

[wbl]

Are these the same proof?

If I let U_i be the interval of length 1/10^(i) centered on f(i), than what I'm saying is pick a different decimal digit to avoid this particular real.

[wbl]

Then we can crank that 2 up to 3 or 4.