hw1 - Modern Analysis, Homework 1 Due June 3, 2010 1. 2. 3....

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Unformatted text preview: Modern Analysis, Homework 1 Due June 3, 2010 1. 2. 3. 4. 5. 6. (1 pt) Let k be a field and x ∈ k be non-zero. Show that 1/(1/x) = x. (3 pts) Rudin, Ch 1, problem 5. (5 pts) Rudin, Ch 1, problem 6. (10 pts) Rudin, Ch 1, problem 7. (2 pts) Rudin, Ch 1, problem 15. Consider the set of infinite decimals, R. That is, expressions of the form ±an an−1 · · · a2 a1 .b1 b2 · · · for some n ∈ N and each ai , bi ∈ {0, 1, 2, · · · , 9}. Also stipulate that if an expression ends with infinitely many 9’s, then that is the same expression with those 9’s replaced by 0’s and the last non 9 entry incremented by one. For instance, 1.299999 · · · = 1.3. Therefore we may assume that any decimal does not end in infinitely many 9’s. Given x ∈ R, define x[n] as the n-th truncation of x, ie change x by replacing each decimal at distance at least n +1 to the right of the decimal point to 0. For instance, if x = 3.1415926 · · · , then x[3] = 3.141 (yeah yeah, infinitely many 0’s, who cares). Given decimals x, y ∈ R, define x < y to mean x[n] < y [n] for some n (here we are using the < on Q). (a) (2 pts) Explain why this is is well defined (ie independent of n) and defines a total order on R. (b) (5 pts) Show that (R, <) satisfies the least-upper-bound property. (c) (Bonus, not necessary) Define addition on R. Then define multiplication. 1 ...
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This note was uploaded on 10/18/2011 for the course MATH S4061X taught by Professor Peters during the Spring '11 term at Columbia.

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