Any hyperreal number greater than 0 and smaller than all positive real numbers is infinitesimal. We know the sum of two infinitesimals is infinitesimal. Let $A$ be the smallest positive real number. Let $dx$ be an infinitesimal. $(A - dx)$ is an infinitesimal as it is less than all positive real numbers
$(A - dx) + dx$ is the sum of two infinitesimals, which under non standard analysis, is infinitesimal. However, I thought that non standard algebra followed the rules of standard algebra, so shouldn't the sum $(A - dx) + dx$ be $A$, a standard real number, and not an infinitesimal? Is there a way around this contradiction? Thanks
Edit: Fixed equation formatting
Okay, thanks guys, so the answer is that $A$ cannot be assumed to exist because it doesn't, as there is always a smaller real number