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Intuitively, we often think of real numbers as existing in one-to-one correspondence with the points on a continuously drawn line, the real number line. One way of expressing the completeness of the real numbers is to say that the real line has no holes.

That is, the values that are “missing” from the rationals—such as, for example, 2 —are present in the real numbers. What do you see as the possible limitations of using this intuitive idea to prove the existence of certain limits in the real numbers? Do you think it is sufficient for the purposes of most students who study calculus to simply accept the existence of these limits without proof?

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For a first calculus course that is taken by "everybody," certainly. Though the students need to be exposed to enough nasty examples that they will know that blind manipulation can lead to incorrect conclusions. – André Nicolas Sep 3 '13 at 17:01
Did you mean to write $\sqrt 2$ instead of $2$ for your example? – Omnomnomnom Sep 3 '13 at 17:03
Well, in higher up math, you don't "prove" these exist, you construct them to exist. So it's not clear how you can prove to a calculus student that they exist. Instead, for calculus, you assume it as an axiom of the real numbers. – Thomas Andrews Sep 3 '13 at 17:06

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