Advanced Calculus: An Introduction to Linear Analysis by Leonard F. Richardson

By Leonard F. Richardson

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Extra resources for Advanced Calculus: An Introduction to Linear Analysis

Example text

The student should not even consider such a route, just as he or she should not substitute a pill for a good meal. We see by playing with the definition of absolute value that Ia I must be equal to either a or -a. This reminds us of what we observed when checking pairs of specific numbers of the same or opposite sign, as explained above. The playfulness appears when we choose to write this as -lal ::; a ::; lal for all a, even though the truth of this double inequality hinges upon a being equal to either the left side or the right side.

A) .!. < E? ,n < E? c) < E? 2 Prove the uniqueness of the additive inverse -a of a. 3 Use the Axiom of Distributivity to prove that aO this to prove that ( -1) ( -1) = 1. 6 Prove: For all a and bin JR, labl = lallbl. 7 Prove: For all a, b, c in JR, Ia- cl :::; Ia- bl + lb- cl. 8 LeU:> 0. Findanumber8 > Osmallenoughsothatla-bl implies Ia - cl < E. 9 < 8andlc-bl < 8 t Prove: For all a and bin JR, llal-lbll:::; Ia- bl. Intuitively, this says that Ia I and lbl cannot be farther apart than a and bare.

13 Let ifn ifn < 100, ~ 100. Prove that Xn converges and find lim Xn. 14 Let Xn = n~l. Prove Xn converges and find the limit. 16 Let Xn = ~. Prove Xn converges and find the limit. 17 Let Xn = n ;;n. Does Xn converge or diverge? Prove your claim. 18 Let Xn = <-Ir+I. Does Xn converge or diverge? Prove your claim. Prove Xn converges and find the limit. 2 t Prove: If Sn ~ tn ~ Un for all n and if both Sn ---+ L and Un ---+ L then L as n ---+ oo as well. 21 Prove or give a counterexample: a) Xn + Yn converges if and only if both Xn and Yn converge.