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21 1. Introductory Ideas The manipulation of inequalities is a crucial technique in analysis. 22) ! 23) j x 2 ~ 0 for all x in IR, and x 2 = 0 if and only if x = O. 24) certainly has far-reaching consequences. First, we record some of the crucial properties of the quadratic expression ax 2 + bx + c, where a i- O. 13 Let a, b, c E IR, with a i- O. (i) ax 2 + bx + c ~ 0 for all x in R if and only if a> 0 and b2 - (ii) the equation ax 2 4ac ::s; OJ + bx + c = 0 has real roots if and only if b2 - 4ac ~ O.

8) of this section we have to make some further definitions. A sequence (an) is said to be bounded above if there exists a real number K such that an ~ K for all n ~ 1, and to be bounded below if there exists a real number L such that an ~ L for all n ~ 1. The sequence (an) is said to be bounded if there exists a real number M such that lanl ~ M for all n ~ 1. 8) that (an) is bounded if and only if it is bounded above and below. A bounded sequence need not be convergent - consider, for example, the sequence ((_l)n+l).

Here the table showing the dependence of Non ds One useful way of looking at the notion of limit is to imagine an argument in which your opponent hands you an € and challenges you to find an appropriate N. 3), and so you can establish that the limit is indeed 1. 4) and let us examine the claim that the limit is 1. 001 for all n > 1. Real Analysis 30 But if your opponent chooses f. 0001 for all n > N, and you are forced to conclude that the limit is not 1. For the limit to be 1 you have to win the argument every time, no matter what f.