The following argument in my previous post was meant facetiously, and is invalid.

"Okay, let's define a validity function v(n), with v(n)=1 meaning valid, and v(n)=0 meaning invalid. We're being told that v(n)=1 for any finite n, but approaches zero as n approaches infinity -- increases without bound. No, "v(n)=1 for any finite n" means that v(n) stays 1 as n approaches infinity.

This may be a counterexample: any finite set of numbers has a maximum. So let S_n be a finite set for each positive n. The union (for n=1 to N) of S_n is also finite (where N is a positive integer). But the union (for n=1 to infinity) of S_n is not finite, and need not have a maximum. So we might say V(N) is 1 but lim (as N->infinity) of V(N) is 0.

Now that I think of it, that's not a genuine counterexample, because the union (for n=1 to infinity) of S_n is not the lim (as N->infinity) of the union (for n=1 to N) of S_n -- at least in the sense of any convergence.

"Okay, let's define a validity function v(n), with v(n)=1 meaning valid, and v(n)=0 meaning invalid. We're being told that v(n)=1 for any finite n, but approaches zero as n approaches infinity -- increases without bound. No, "v(n)=1 for any finite n" means that v(n) stays 1 as n approaches infinity.

This may be a counterexample: any finite set of numbers has a maximum. So let S_n be a finite set for each positive n. The union (for n=1 to N) of S_n is also finite (where N is a positive integer). But the union (for n=1 to infinity) of S_n is not finite, and need not have a maximum. So we might say V(N) is 1 but lim (as N->infinity) of V(N) is 0.

Now that I think of it, that's not a genuine counterexample, because the union (for n=1 to infinity) of S_n is not the lim (as N->infinity) of the union (for n=1 to N) of S_n -- at least in the sense of any convergence.

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