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Failed to connect to MySQL: (1203) User howardkn already has more than 'max_user_connections' active connectionsPath: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: wij Newsgroups: comp.theory Subject: Re: Cantor Diagonal Proof Date: Fri, 11 Apr 2025 19:52:20 +0800 Organization: A noiseless patient Spider Lines: 85 Message-ID: References: <1ab7fe6b234496769adde06995790eebb827756e.camel@gmail.com> <60cbb326c7d65b1bbd9451319bd07721c76d307f.camel@gmail.com> <3e5a55b834962635ca7ecf428d074fba771a07f8.camel@gmail.com> <878qo74kbl.fsf@nosuchdomain.example.com> <875xjbt041.fsf@nosuchdomain.example.com> MIME-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Injection-Date: Fri, 11 Apr 2025 13:52:21 +0200 (CEST) Injection-Info: dont-email.me; posting-host="05950bb31754b8261a44aa641f85531a"; logging-data="1475815"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18nYY5j+K1gFyP94dMS79qn" User-Agent: Evolution 3.54.3 (3.54.3-1.fc41) Cancel-Lock: sha1:RTFVIFYgYpwu/frmi1UHskE+1QE= In-Reply-To: <875xjbt041.fsf@nosuchdomain.example.com> Bytes: 4697 On Fri, 2025-04-11 at 04:21 -0700, Keith Thompson wrote: > wij writes: > > On Thu, 2025-04-10 at 17:23 -0700, Keith Thompson wrote: > > > wij writes: > > > [...] > > > > "lim(x->c) f(x)=3DL" means the limit of f approaching c is L, not f= (c)=3DL 'eventually'. > > > > f at c is not defined (handled) in limit. > > >=20 > > > Correct. > > >=20 > > > > lim 0.333...=3D1/3=C2=A0=C2=A0=C2=A0 ... The *limit* is 1/3, not 0.= 333...=3D1/3 > > > > 0.3+0.33+0.333+...=C2=A0 ... The sequence converges to 1/3 > > > > =CE=A3(n=3D1,=E2=88=9E) 3/10^n=C2=A0=C2=A0=C2=A0=C2=A0 ... The sum = converges to 1/3 (or you can use lim) > > >=20 > > > The limit as the number of 3s increases without bound *is exactly wha= t > > > we mean* by the notation "0.333...".=C2=A0 Once you understand that, = it's > > > obvious that 0.333... is exactly equal to 1/3, and that 0.333... is a > > > rational number. > >=20 > > You agree "f at c is not defined (handled) in limit", yet, on the other= hand > > ASSERTING 0.333... is 'exactly' 1/3 from limit? Are you nut? > >=20 > > As usual, you need to prove what you say. Or you are just showing yours= elf=20 > > another olcott, just blink belief, nothing else. >=20 > Keep the insults to yourself.=C2=A0 Last warning. I still think 'nut' is a common word, at least a terse word for people sayi= ng one thing and doing the other (or a liar more appropriate?) > My assertion is simply about what the "..." notation means. >=20 > Do you agree that the limit of 0.3, 0.33, 0.333, as the number of 3s > increases without bound, is exactly 1/3?=C2=A0 (You said so above.) Increases without bound -> yes is exactly 1/3 -> no such logic > What exactly do you think the notation "0.333..." means?=C2=A0 I and many > others use that notation to mean the limit, which you agree is exactly > 1/3.=C2=A0=C2=A0 Is this a lie? I have always consistently claiming "repeating decimals are = irrational". > Why do you object to the use of that particular notation for that > particular concept? >=20 Illogic > Do you have any mathematical argument that isn't purely about notation > that you dislike? ??? > Would, say "0.333<=E2=88=9E>" be clearer?=C2=A0 Could you agree that that= refers to > the limit and gives a result that's exactly 1/3?=C2=A0=C2=A0 0.333... approaches 1/3 --> no problem. 0.333... equals exactly to 1/3 --> no way (I have provided proofs and you d= on't). > If so, why do you > object to "..." but not to "<=E2=88=9E>" as a symbol for the limit?=C2=A0= (Note that > "..." is easier to type, unless you happen to have an =E2=88=9E key your > keyboard.) Who say I object the use of "..."? As said, it is 'the limit', not exactly equal (as explained) As usual, you still only have irrelevant garbage talk, no valid logic proof= .. If so, I can choose to stop responding.