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New PDF release: A course in real analysis

By Hugo D. Junghenn

ISBN-10: 148221928X

ISBN-13: 9781482219289

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There exists a nonempty subset P of R, closed under addition and multiplication, such that for each x ∈ R exactly one of the following holds: x ∈ P, −x ∈ P, or x = 0. The last part of the axiom is known as the trichotomy property. A real number x is called positive if x ∈ P and negative if −x ∈ P. 1 Definition. Let a and b be real numbers. If b − a ∈ P, we write a < b or b > a and say that a is less than b or that b is greater than a. 2 Proposition. The order relation < on R has the following properties: (a) a < b iff −a > −b (reflection property).

What if x < −1 < y? 6. If 0 < x < y and n ∈ N, then (a)S 0 < y n − xn ≤ n(y − x)y n−1 , 7. If x > 1, m, n ∈ N, and (b) ny + 1 (n + 1)y + 1 < . nx + 1 (n + 1)x + 1 x−1 m < < 1, then n > x. S If a < b and 0 < t < 1, then a < ta + (1 − t)b < b. In particular, a < (a + b)/2 < b. 9. x2 + y 2 + axy ≥ 0 for all x, y ∈ R iff |a| ≤ 2. S If a ≤ b + x for every x > 0, then a ≤ b. The Real Number System 11 11. If 0 < a ≤ bx for every x > 1, then a ≤ b. 12. If a/x ≤ x + 1 for every x > 0, then a ≤ 0. 13. For all x, y, z, w ∈ R, (a) 2xy ≤ x2 + y 2 .

9 Example. If r > 1, then rn /n → +∞. 6: Given M > 0 there exists N ∈ N such that n/rn < 1/M , hence rn /n > M , for all n ≥ N. 10 Example. If r > 0, then an := rn n! → +∞. Indeed, since an = rn → +∞, an−1 there exists N ∈ N such that an > 2an−1 , for all n > N . Iterating, we see that an > 2k an−k ≥ kan−k , so taking k = n − N we have an > (n − N )aN for all n > N . Since limn (n − N )aN = +∞ (Archimedean principle), the assertion follows. 12). 11 Theorem. Let {an } and {bn } be sequences in R.

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A course in real analysis by Hugo D. Junghenn


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