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Aug
26
comment Proof that Pi is constant (the same for all circles), without using limits
To me, this proof is a typical proof which uses limits and avoids the mere use of the words “limits” or “converge.” I would say that you prove some property about Cauchy sequences converging to π(C_1) and π(C_2), but even if you avoid the word “converging” by saying that the Cauchy sequence is the very definition of the real number, a proof working on the defining Cauchy sequences sounds like a proof using limits to me.
Aug
25
comment What does “only” mean?
@Américo: Thanks for the reference. I see that the word “provided” sometimes mean “only if” or “if and only if” in English just as in the example given in Wiktionary. I do not think that this usage is common in mathematical context, but I may be wrong.
Aug
25
comment What does “only” mean?
@Américo Tavares: That is the other way around. “A only if B” (where A and B are clauses) means “B if A,” and equivalent to “B provided that A.” (I am not a native speaker of English, either.)
Aug
24
comment Proof that Pi is constant (the same for all circles), without using limits
@Niel: I do not know whether similarity is useful in non-Euclidean geometry. Probably I should have restricted my claim to mathematical facts about Euclidean geometry (or to mathematical facts about the length of circumference!).
Aug
24
comment Proof that Pi is constant (the same for all circles), without using limits
@Niel de Beaudrap: I would say that it is part of the “intuitive” understanding of the notion of similarity. This can be explained by noting that mathematical facts are independent of the choice of unit length. But I am no expert of mathematical education, and I am not too sure what is intuitive and what is not.
Aug
24
revised Proof that Pi is constant (the same for all circles), without using limits
added 210 characters in body
Aug
24
answered Proof that Pi is constant (the same for all circles), without using limits
Aug
14
awarded  Supporter
Aug
13
awarded  Teacher
Aug
13
revised a provable upper bound on the summation
fixed typo, added some comment
Aug
13
awarded  Editor
Aug
13
revised a provable upper bound on the summation
better formatting
Aug
13
answered a provable upper bound on the summation