By Halsey Royden

ISBN-10: 0024041513

ISBN-13: 9780024041517

This is often the vintage introductory graduate textual content. center of the booklet is degree concept and Lebesque integration.

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Additional resources for Real Analysis (3rd Edition)

Example text

E) Use the previous two parts of this exercise to show that if {xk } is bounded and x is its one and only accumulation point, then xk → x. 46 CHAPTER 2. SEQUENCES AND THEIR LIMITS n 31. Let {xk }∞ k=1 be a sequence of real numbers, and let Sn := k=1 xk be ∞ the sequence of partial sums for the series k=1 xk . 10). , there is some α > 0 such that xk = α for all k) then Sn → ∞. (b) Show that if the sequence is (not necessarily constant, but) bounded below by some strictly positive number (xk ≥ α > 0 for some α), then Sn → ∞.

We have just established that if 2 is rational, then there is an √ integer q which is odd and even at the same time. That’s absurd! 3 √ OK, that’s settled. 2 is definitely not rational. How, then, do we locate it on the number line? The best we can do is to devise a scheme that lets us √ position 2 relative to the rationals (whose location we can get exactly). We shall consider two schemes to do this. √ The first—and most familiar one—is to find a decimal expansion for 2. ) we conclude from Exercise 11√that 2 lies between 1 and 2.

A finite sequence of n real numbers is essentially an n-tuple, but to consider limits we need infinite sequences, so whenever we say “sequence”, we mean that it is infinite unless it is explicitly called finite. Some sequences are understood once we establish a pattern with the first few terms. For example, 1, 2, 3, 4, . . is normally recognized as the sequence of positive integers, while 1 1 1 1 1 , , , , ,... 1 2 4 8 16 is understood to be the reciprocals of powers of 2. This is, however, an unreliable method; we prefer a specification that explicitly tells us how to find the value of each number in the sequence.