By Arlen Brown

As its name shows, this ebook is meant to function a textbook for an introductory path in mathematical research. In initial shape the booklet has been utilized in this manner on the collage of Michigan, Indiana collage, and Texas A&M college, and has proved serviceable. as well as its fundamental goal as a textbook for a proper direction, although, it's the authors' desire that this booklet also will turn out of worth to readers drawn to learning mathematical research on their lonesome. certainly, we think the wealth and diversity of examples and workouts may be specifically conducive to this finish. A be aware on must haves. With what mathematical historical past may perhaps a potential reader wish to learn from the examine of this ebook? Our con scious purpose in writing it used to be to handle the wishes of a starting graduate pupil in arithmetic, or, to place concerns just a little in a different way, a pupil who has accomplished an undergraduate software with a arithmetic ma jor. however, the booklet is especially mostly self-contained and will accordingly be available to a reduce classman whose curiosity in mathematical research has already been awakened.

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**Sample text**

If x, x' E IC(T) and a, (3 are scalars, then T(ax+(3x') = a·O+(3·O = 0, so ax + (3x' E IC(T). Similarly, if y, y' E 'R(T), and if y = Tx, y' = Tx', and a, (3 are scalars, then ay + (3y' = T( ax + (3x') E 'R(T). Moreover, it is clear that IC(T) = (0) if T is one-t~one.

Thus the set R of all reciprocals of the natural numbers is a subset of JR with inf R = O. Example B. The function

providing an order isomorphism of all of IR onto ( -1, +1) (cf. Problems G and L). 2. If m and n are natural numbers, then their sum m + n and their product mn are also natural numbers.

J. The square t 2 = tt of every real number is nonnegative, while the square of every nonzero real number is positive. In particular, 1 > 0, and s < s + 1 for every real number s. K. If 8 is a positive [negative] real number, then l/s is also positive [negative]. If s and t are positive real numbers s. ch that s < t, then l/t < 1/ s. L. For any real number s the absolute value lsi is nonnegative and equal to I - sl, while lsi = 0 if and only if s = O. For any real number 8 we have s :S lsi and -s :S lsi.