A common mistake among students just learning the calculus is a confusion between a quantity being allowed to approach infinity and actual instances of infinity. For example: It is incorrect to say that 0.333... is not equal to 1/3. The common error is to assert that "as the threes go on and on, you get closer and closer to one third, but you can't actually get there since the threes would have to be infinite." The perpetrator of this error is assuming that the threes are not already infinite. However, the expression 0.333... indicates that an infinite number of threes already exist, as implied by the ellipsis. 0.333... does not need an observation or anyone's permission to have an infinite number of threes. It is neither necessary nor proper to speak of the value "approaching" some other value. The value of 0.333... is concrete. If one is thinking of a number that begins 0.333 that ends at some arbitrary point, and imagining adding 3s one by one, that person is not thinking of the number 0.333... at all. It is only proper to speak of the value of a formula, sequence, or series approaching a given value in the context of limits. "0.333..." does not require or imply that the use of a limit is appropriate. It is proper to say that 0.333... is exactly equal to 1/3. By extension, 0.666... does not have a seven "at the end", nor is 0.999... "almost but not quite equal to one." Novices to the field of epsilontics will claim that 0.999... "can be made as close to 1 as you like, but will never equal 1." They are incorrect. The value of 0.999... is well-defined, equal to 1, and not subject to change due to observation. Hope that helps some of you. It's a small bit of how I explain infinity to my pre-cal students. And my favorite paradox is not so much a paradox as an exercise in counterintuitive thought: At exactly 7 am one morning, a hiker begins to ascend a mountain trail. The hiker does not necessarily move at a constant speed, and may stop to rest, but never backtracks along the path. Sometime later, the hiker reaches the top of the mountain and sets up camp for the night. At exactly 7am the following morning, the hiker begins to descend the mountain by the exact same path. Again, the hiker may travel at any speed, even stopping, but does not backtrack. The question: is it impossible, possible, or necessary that there should exist one or more points along the trail that the hiker passes at exactly the same time of day coming down the mountain on day 2 as going up the mountain on day 1? If it is possible or necessary, how many points meet this requirement? Good luck with the reasoning. Mensch