Hello all! In the past few months I've stumbled upon an issue that has played games with my mind. I feel I need some help to solve this, as I've tried various other sources and remain without answers. Firstly, I was confronted with a mathematical proof which states that 0.9~ (to infinity) = 1. The proof is simply: x = 0.9~ 10x = 9.9~ 9x = 9 x = 1 This bugs me, for from a purely logical standpoint it seems ludicrous to claim that two different numerical representations are representative of the same quantitative value. As a result of studying this issue I have also run into a problem with my whole perception of reality. All my life, I have considered mathematics to be an ultimate truth which has certain affinity with reality. Thus, what is proven in mathematics, has for me been enough proof for such a truth in reality. In mathematics, infinity exists and is essential to the functionality of the number system. In reality, however, I struggle to find an example of infinity which can be tied directly to its existence in mathematics. To further this conundrum, I have realized that, should there be infinite distance between any two given points in reality, there can be no points to begin with, for a point which is endlessly broken down ad infinitum has no end. My entire perception of time and space has been confused, and I'd like some help to understand why. Thank you in advance.