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It's still not true. No no sense is it true. It is true that 0.999... = lim[eps->0] (1 - eps), but it is ALSO true that lim[eps->0] (1 - eps) = 1. That's how li
by OskarS 7y ago
It's still not true. No no sense is it true. It is true that 0.999... = lim[eps->0] (1 - eps), but it is ALSO true that lim[eps->0] (1 - eps) = 1. That's how limits work.
If you accept both of those two (which you should, because they're correct), then since equality is transitive, 0.999... = 1. Therefore it is not less than 1, it is equal to 1.
- amelius 7y agoI think you should look more closely at what I did with the order of operations, and the fact that "<" now is part of the expression acted over by the limit.
- danharaj 7y agoIt is not reasonable to interpret the notation that way. For one, variable binding becomes incoherent.
- amelius 7y agoI think it is perfectly reasonable for two reasons. One is that (when it works) the notation becomes more intuitive, as in 0.9999... < 1 And the second reason is that mixing implicit limits (or series or infinity) and numbers is a mathematical hack, and when the notation doesn't work it is a clear indication that something is wrong.
- OskarS 7y agoLimits 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.
- GolDDranks 7y agoYou know, inequalities can be interpreted as boolean functions. The function under the limit produces consistently true as you approach the limit. (Of course one-sided one; the case where epsilon is getting smaller). It isn't a valid argument for 0.999... != 1 but an interesting one nevertheless.