String of recent killings linked to Bay Area 'Zizians'

In that case, you would've reached their respective limits, and you're back to adding one of those limits into itself an other-limit number of times. If sₙ · tₙ ⟶ 1, then that only holds true if tₙ hasn't actually reached 0.

> Thus we can either dispense with the cited established theorems of analysis used to deduce that sₙ · tₙ ⟶ 1

You don't need to do that. 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.

> It might be the case that Σ(∞, i = 1) 0 = 0, but you can't extend this to conclude +∞ · 0 = 0 in general.

Sure you can, unless you've got some other definition of multiplication that's impossible to express as self-summation.

Even if you go with the alternative definition of multiplication as a scaling operation (wherein you're computing m × n by taking the slope from (x=0,y=0) to (x=1,y=m) and then looking up y where x=n), if m is zero then the line being drawn never stops being vertical, and if n is zero then you never leave (0,0) in the first place. Doesn't matter if the other factor is infinitely far in either the x or y axis; you're still ending up with zero no matter how hard you try and fight it.

> Lots of intuitions from informal mathematics and even calculus start to break down once you examine the lower-level "machine code" of proof and analysis, especially once you start talking about concepts like infinity.

Sure, but in this case, it's the intuition that multiplying something by its inverse (a.k.a. dividing something by itself) is always 1 that breaks down, not the above-verifiable and inescapable fact that multiplying something by zero is always zero. 0 ÷ 0 = n looks like it should correct for any value of n (incl. n = 1), since multiplying both sides by zero to eliminate that divide-by-zero will always produce a correct equation, but since m ÷ n ≡ m × (1/n), if m is zero then anything on the RHS must be zero, because of that inescapable nature of nothingness - thus, 0 ÷ 0 = 0 × (1/0) = 0, with all other possible alternatives having been rendered impossible.

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.