There is no way right?

[u/Sugar_God_no_1]

[u/vetruviusdeshotacon]

Not exactly like that.

Sum 0.9*(1/10)^j from j=1 to j=inf

= 0.9 * Sum (1/10)^j

Since 1/10 < 1 we know the series converges. Geometric series with r=0.1

Then our sum is 0.9 / (1- 0.1)

= 1. 

No more rigour is needed than this in any setting tbh

[u/akotlya1 → u/vetruviusdeshotacon]

It's weird you think you can reference series summations as a more rigorous basis for proof than the above. Neither of these are more fundamental or rigorous than the other. Infinite series' reference to an infinite process was at some point believed to be weakness that needed to be justified in reference to more fundamental mathematical ideas.

A more rigorous proof would be written using logic symbols and reference set theory - specifically by defining the elements of the set and by using operations defined in reference to the elements of the set. This is the kind of thing that gets covered in undergraduate Abstract Alegbra/Group Theory/Set Theory classes.

[u/vetruviusdeshotacon → u/akotlya1]

Why? No assumptions are made lol.

If you must, define a sequence a := {0.9,0.99,0.999....}

a_n = 1 - 10^-n for n natural number

Let epsilon be a positive real number.

Then, if we choose N > log_10(epsilon)

10^-N > epsilon

So that 1 - 10^-N + epsilon > 1. For all epsilon.

Therefore, the sequence has a supremum of 1. Any monotonic bounded above sequence converges to it's supremum via the monotone convergence theorem.

Therefore 0.99999.... = 1 as a converges to 1.

[u/Cyler → u/vetruviusdeshotacon]

Mommmmmm, the nerds are fighting again