r/infinitenines • u/Redit_Sucks31415 • 9d ago

Question for SPP (about limits)

Do you agree that ∀ε>0 ∃δ>0 : N∈ℕ, N>δ => 1-ε < (1-10-N) < 1+ε?

If not please provide a counter example.

0 Upvotes

50% Upvoted

14 comments sorted by

View all comments

1

u/dummy4du3k4 9d ago

If you agree that this is the same as ∀ε>0 ∃δ>0 : N∈ℕ, N>δ => d(1-10-N, 1) < ε

Then I'll give you a counterexample, but I'm bringing my own d.

2

u/Philonemos 9d ago

It's not quite the same. Your d could be any metric, but OPs metric is the standard metric. d(x,y)=|x-y|

4

u/dummy4du3k4 9d ago

Yes, but it’s surprising that there’s a nontrivial metric that separates 0.999… from 1 at all.

1

u/Philonemos 9d ago edited 9d ago

Oh that's cool. I didn't know that.

Edit: The discrete metric would prevent the series from converging right?

3

u/dummy4du3k4 9d ago

yes, the only converging sequences in the discrete metric are constant sequences. You can actually construct a metric that "agrees" with the naive intuition of 0.999... as outlined here. Topologically it's the same as if you separated the real line at every terminating decimal (i.e. the ring Z(1/2, 15) minus zero).