> you're back to adding one of those limits into itself an other-limit number of times.
> some other definition of multiplication that's impossible to express as self-summation.
Ok. What happens if I multiply a number by pi? What does it mean to add something to itself, pi times?
> If sₙ · tₙ ⟶ 1, then that only holds true if tₙ hasn't actually reached 0.
I mean... it is in fact the case that tₙ never actually reaches zero; otherwise, if 1/n = 0 for some n, then by multiplying both sides by n we obtain 1 = 0.
What's meant by tₙ ⟶ 0 is that any neighborhood centered about 0 of any radius (call the radius "epsilon") always contains at least one point from the sequence {tₙ}.
To hammer the point that sₙ · tₙ ⟶ 1 home, and since you are fond of using a computer to perform arithmetic (note: not prove mathematical statements), here's what computers have to say about the limit of n · (1/n): https://www.wolframalpha.com/input?i=limit+as+n-%3Einfinity+...
> You just need to accept that zero is just as much of a mathematical special case as infinity - unsurprisingly, since it's the inverse of infinity and vice versa.
> the inverse of infinity
You again throw around words like "inverse" whose meaning you don't understand. Do you mean a multiplicative inverse, where a number and its multiplicative inverse yield the multiplicative identity, in which case +∞ · 0 = 1? Or an additive inverse that yields the additive identity, in which case +∞ + 0 = 0? Or some other pseudomathematical definition of "inverse" pulled out of a hat, like your definitions of +∞ · 0?
> if m is zero then the line being drawn never stops being vertical
Drawing pictures is different from putting together a formal, airtight proof in first-order logic that can be (in principle) machine-verified. Maybe I'll make an exception for compass-and-straightedge proofs, but that's not what you're presenting here.
Rudin was published in 1953, there are probably very good reasons for why this text has withstood refutation for over 70 years. Maybe you can rise to the task; publish a paper with your novel number system in which +∞ · 0 = 0 and 0 ÷ 0 = 0 and wait for your Fields Medal in the mail. Maybe you can collaborate with Terrence Howard and get a spot on Joe Rogan.
> Ok. What happens if I multiply a number by pi? What does it mean to add something to itself, pi times?
You add it to itself 3 times, then shift the decimal point and repeat with 1, then shift the decimal point and repeat with 4, and so on with each digit of π. 1 × π = 1 + 1 + 1 + 0.1 + 0.01 + 0.01 + 0.01 + 0.01 + 0.001 + 0.0001 + 0.0001 + 0.0001 + 0.0001 + 0.0001 and so on forever.
> To hammer the point that sₙ · tₙ ⟶ 1 home
That point doesn't need hammered. sₙ · tₙ ⟶ 1 can absolutely be true when you haven't yet reached zero. That doesn't mean it's true in the event that you do indeed manage to reach zero. It indeed can't be true in the event that you do indeed reach zero, because n × 0 = 0 for all values of n.
> Do you mean a multiplicative inverse, where a number and its multiplicative inverse yield the multiplicative identity, in which case +∞ · 0 = 1?
You obviously already know that's what I meant, since that's exactly what I described further down - including how ∞ × 0 ≠ 1 because the multiplicative identity breaks down when one of the factors is zero, specifically because having zero of something will always produce zero no matter what that something is.
> Or some other pseudomathematical definition of "inverse" pulled out of a hat, like your definitions of +∞ · 0?
If you're seriously calling multiplication-as-summation pseudomathematics, then you're in no position to assess whether or not I "don't understand" the meanings of words.
I've been nothing but civil toward you, and you've been nothing but condescending toward me. That normally wouldn't be a problem (condescension is par for the course on the Internet), but if you're going to be condescending, the least you can do is not be blatantly wrong in the process.
> Drawing pictures is different from putting together a formal, airtight proof in first-order logic that can be (in principle) machine-verified. Maybe I'll make an exception for compass-and-straightedge proofs, but that's not what you're presenting here.
That's exactly what I'm presenting here (since apparently you believe adding numbers together is a spook). You don't even need a concept of numbers to see plain as day that any multiplication wherein one of the factors is zero will always be zero.
> Rudin was published in 1953, there are probably very good reasons for why this text has withstood refutation for over 70 years. Maybe you can rise to the task; publish a paper with your novel number system in which +∞ · 0 = 0 and 0 ÷ 0 = 0 and wait for your Fields Medal in the mail. Maybe you can collaborate with Terrence Howard and get a spot on Joe Rogan.
You know what? Maybe I will. And I'm willing to bet you'll find some other pedantic reason to be a condescending prick when that happens.
Last word's yours if you want it. I have better things to do than argue with people engaging in bad faith.
There's no need for me to continue engaging you with formal mathematical arguments when you reply with the mathematical equivalent of climate change denialism or vaccine conspiracy theory and uneducated statements that are "not even wrong" [0], so instead I will just refer you to expert opinions on the topic; though at this point I doubt that your level of mathematical literacy is sufficient to understand any of this subject matter.
> some other definition of multiplication that's impossible to express as self-summation.
Ok. What happens if I multiply a number by pi? What does it mean to add something to itself, pi times?
> If sₙ · tₙ ⟶ 1, then that only holds true if tₙ hasn't actually reached 0.
I mean... it is in fact the case that tₙ never actually reaches zero; otherwise, if 1/n = 0 for some n, then by multiplying both sides by n we obtain 1 = 0.
What's meant by tₙ ⟶ 0 is that any neighborhood centered about 0 of any radius (call the radius "epsilon") always contains at least one point from the sequence {tₙ}.
To hammer the point that sₙ · tₙ ⟶ 1 home, and since you are fond of using a computer to perform arithmetic (note: not prove mathematical statements), here's what computers have to say about the limit of n · (1/n): https://www.wolframalpha.com/input?i=limit+as+n-%3Einfinity+...
> You just need to accept that zero is just as much of a mathematical special case as infinity - unsurprisingly, since it's the inverse of infinity and vice versa.
> the inverse of infinity
You again throw around words like "inverse" whose meaning you don't understand. Do you mean a multiplicative inverse, where a number and its multiplicative inverse yield the multiplicative identity, in which case +∞ · 0 = 1? Or an additive inverse that yields the additive identity, in which case +∞ + 0 = 0? Or some other pseudomathematical definition of "inverse" pulled out of a hat, like your definitions of +∞ · 0?
> if m is zero then the line being drawn never stops being vertical
Drawing pictures is different from putting together a formal, airtight proof in first-order logic that can be (in principle) machine-verified. Maybe I'll make an exception for compass-and-straightedge proofs, but that's not what you're presenting here.
Rudin was published in 1953, there are probably very good reasons for why this text has withstood refutation for over 70 years. Maybe you can rise to the task; publish a paper with your novel number system in which +∞ · 0 = 0 and 0 ÷ 0 = 0 and wait for your Fields Medal in the mail. Maybe you can collaborate with Terrence Howard and get a spot on Joe Rogan.