That equation proves very little. if you want to get technical, I guess we can (even though I am lazy).
BY DEFINITION, $$ 0.111... = \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{10^i}$$.
Ok? Also, BY DEFINITION, $$\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{10^i}$$ is equal to the limit of the sequence of partial sums $$(s_1, s_2, ... )$$ where $$s_n = \sum_{i=1}^{n} \frac{1}{10^i}$$. Does EVERYONE see this? We are NOT summing infinite terms by this definition... people seem to think that because summing an infinite number of terms is physically impossible we cannot describe what such a sum may look like.
OK, we notice two things people. The first... $$s_n = \sum_{i=1}^{n} \frac{1}{10^i}$$ is an non-decreasing. What does this mean? $$s_{n+1} > s_{n} \ for \ all \ n \geq 1$$. So, clearly... $$s_1 < s_2 < ... < s_k < ... $$.
What's the second thing we notice?!?!? $$For \ all \ s_n, s_n < 1$$.
These two points are very significant. Why? Think about the real world for a second... if you fill a balloon with helium it floats, right? Letting go of the balloon and we see that its height is NON-DECREASING (in other words, its height is strictly increasing). If you let this balloon float up and up and up in a building with a roof, it's going to stop floating when it hits the roof.
This sequence $$s_n$$ is just like this balloon. It increases... but it's bounded by a "roof." We know that this room is at MOST 1. However, we can show that 1 is the shortest roof for this sequence (least upper bound, people). How?
I'll explain later. My class starts in a few minutes.