By Terence Tao
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Extra resources for Analysis I (Volume 1)
Using an advanced fact to prove a more elementary fact, and then later using the elementary fact to prove the advanced fact). Also, this exercise will be an excellent way to affirm the foundations of your mathematical knowledge. Furthermore, practicing your proofs and abstract thinking here 16 2. The natural numbers will be invaluable when we move on to more advanced concepts, such as real numbers, functions, sequences and series, differentials and integrals, and so forth. In short, the results here may seem trivial, but the journey is much more important than the destination, for now.
5 should technically be called an axiom schema rather than an axiom - it is a template for producing an (infinite) number of axioms, rather than being a single axiom in its own right. ) The informal intuition behind this axiom is the following. Suppose P(n) is such that P(O) is true, and such that whenever P(n) is true, then P(n++) is true. Then since P(O) is true, P(O++) = P(1) is true. Since P(1) is true, P(1 ++) = P(2) is true. Repeating this indefinitely, we see that P(O), P(1), P(2), P(3), etc.
Does not go to zero ) . So the hm1t hmx--+0 1 diverges. One might then conclude using L'Hopital's rule that 2 . ( -4) limx--+0 _x smx x also diverges; however we can clearly rewrite this limit as limx--+0 xsin(x- 4), which goes to zero when x---... 0 by the squeeze test again. 5), but it still requires some care when applied. 13 (Limits and lengths). When you learn about integration and how it relates to the area under a curve, you were probably presented with some picture in which the area under the curve was approximated by a bunch of rectangles, whose area was given by a Riemann sum, and then one somehow "took limits" to replace that Riemann sum with an integral, which then presumably matched the actual area under the curve.