Limits are defined for functions and "1 - eps < 1" is not a function, it's an inequality. But even if we pretended that limits were defined for inequalities, th

Limits are defined for functions and "1 - eps < 1" is not a function, it's an inequality. But even if we pretended that limits were defined for inequalities, there only reasonable interpretation of it is that: lim[eps->0]( 1 - eps < 1) is equal to 1 - 0 < 1 which is obviously false. Again though, limits are defined for functions, not inequalities. If you have the limit lim[eps->0] (x - eps) Then it is equal to x - 0 which is equal to x. You're simply wrong here. You can read that wikipedia article if you don't trust me, or you can watch any of the thousands of youtube videos of mathematicians explaining this to you. This is one of those things that is hard to do. Once you have an idea in your head that you're sure is right, it is incredibly difficult to dislodge it. It takes an enormous amount of humility and intellectual flexibility. But it is healthy and good to do it, every time you do it, you become a better person for it.