Real Analysis walter rudin solutions pdf

We introduce the set of real numbers in an informal way. Once this is done,

the most important property of the real number system, known as the least

upper bound property, is introduced. We assume that you know the set N of

natural numbers, the set Z of integers, and the set Q of rational numbers. You

know that N ⊂ Z ⊂ Q. You also know the arithmetic operations, addition and

multiplication of two natural numbers, integers, and rational numbers. You also

know the order relation m < n between two integers, more generally between two

rational numbers.

Sequences arise naturally when we want to approximate quantities. For in-

stance, when wish to use decimal expansion for the rational number 1/3 we get a

sequence 0.3, 0.33, 0.333, . . .. We also understand that each term is approximately

equal to 1/3 up to certain level of accuracy. What do we mean by this? If we want

the difference between 1/3 and the approximation to be less than, say, 10−3

, we

may take any one of the decimal numbers 0.

3 . . . 3 where n > 3. Or, if we want

to use decimal expansion for √

2, we look at

1.4, 1.41, 1.414, 1.4142, 1.41421, . . . , 1.4142135623730950488016887242 . . . .

The idea of a sequence (xn) and its convergence to x ∈ R is another way of

saying that we give a sequence of approximations xn to x in such a way that if

one prescribes a level of accuracy, we may ask him to take any xn after some

N-th term onward. Each such will be near to x to a desired level of accuracy.

It is our considered opinion that students who master this chapter will begin

to appreciate analysis and the way the proofs are considered.