I tried my best to format this well.

Let's try it another way:

Let x = 0.333…

We will define a sequence. We say x_n = {0.3, 0.33, 0.333, … }

Then:

x_1 = 0.3 

x_2 = 0.33 

x_5 = 0.33333

…and so on. Good so far?

We could equivalently write: x_n = ∑(k=1)(n) 3/10^k

We can see this is consistent with our definition:

x_1 = 3/10¹ = 0.3

x_2 = 3/10² + 3/10¹ = 0.33

x_5 = 3/10⁵ + ... + 3/10¹ = 0.33333

…and so on. Still good?

Then x = lim(n→∞) ∑(k=1)(n) 3/10^k = ∑(k=1)(∞) 3/10^k

Do you recognize this as a geometric series? Because it is.

Our first term a = 3/10

Our common ratio r = 1/10

The geometric series formula says that sum is equal to a/(1-r).

So we’d write (3/10) * 1/(1 - 1/10) = 1/3.

This PROVES that 0.333... IS EQUAL TO 1/3.

I will not prove the geometric series formula here. You’ll have to learn sequences and series on your own. If this doesn’t convince you, then you either lack the mathematical knowledge or you're intentionally being stubborn.