>be me, /sci/ chad, watching you mouthbreathers bicker over 0.999... and staircase paradoxes like you just discovered math yesterday.
half you plebs: "duh 0.999... = 1, limits go brrr, no infinitesimals in reals, get rekt"
other half: "nah it's [math]1 - \frac{1}{\infty}[/math], wake up sheeple, feel the gap"
Lmao you brainlets are arguing past each other because you're playing in different sandboxes with unspoken axioms. Peak /sci/ bait thread, here we go.
In your normie-tier [math]\mathbb{R}[/math] with Euclidean norm (L2), yeah, 0.999... is 1. It's literally [math]\lim_{n\to\infty} (1 - 10^{-n})[/math]. In that basic system, nonzero infinitesimals don't exist. The gap is zero. Done deal.
But step into the hyperreals ([math]^*\mathbb{R}[/math]), and you get infinite hyperintegers H where a decimal with H nines is exactly 1-10^-H. That 10^-H? A legit positive infinitesimal, your caveman "[math]\frac{1}{\infty}[/math]". In this framework, 0.999... being just shy of 1 is correct. It's 1 minus a tiny epsilon that ain't zero. Both sides are right, you walnuts, just in different universes. Pick one and stop crying.
Same with the staircase vs diagonal autism. In Euclidean L2 norm, the diagonal is [math]\sqrt{a^2 + b^2}[/math], staircase is a+b. Not equal. But flip to Manhattan L1 norm, and bam: the "straight line" from (0,0) to (a,b) is a+b. Staircase and "diagonal" are the same length. It all depends on your metric. Thinking Euclidean is the only truth? Cringe.
So keep smashing rocks in basic [math]\mathbb{R}[/math], yelling "equals 1!", or level up and see why the other side feels right. The fight ain't about math, it's you smoothbrains not knowing your axioms. Absolute kino.
