Appendix i-
Proofs of basic theorems on limits
This appendix contains proofs of the basic theorems on limits which
were given without proof in Section 3.2. Persons having competence
in mathematical analysis must know these theorems and be able to give
their proofs as thoroughly and as expertly as competent violin and piano
players know and can play their scales. Individuals having nontrivial
mathematical ambitions must therefore study the material of this appen-
dix more than once. Most of the proofs depend upon the fundamental
fact that if x and y are numbers, then
(1) Ix + yl : Ixl + IyI, Ix - yI = IxI + Iyl
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