If r is rational (r not equal to 0) and x is irrational

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Reference no: EM13741544

1) Rudin, Ch. 1, Problem 1: If r is rational (r not equal to 0) and x is irrational, prove that r+x and rx are irrational.

2) If (a_n), (b_n) are rational sequences, and lim a_n = a, lim b_n = b, show that: lim(a_n+b_n) = a+b, and lim(a_n b_n) = ab.

3) Show: If a_n converges to a, any subsequence (a_(n_k)) of (a_n) converges to a.

4) Let (b_n) be a Cauchy sequence of rational numbers such that b_n is not zero for any n. Show that if (b_n) does not converge to 0, then (b_n) has a multiplicative inverse in the set of Cauchy sequences.

5) Show that between any two distinct real numbers there are infinitely many rational numbers.

Reference no: EM13741544

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If r is rational (r not equal to 0) and x is irrational : 1) Rudin, Ch. 1, Problem 1: If r is rational (r not equal to 0) and x is irrational, prove that r+x and rx are irrational.2) If (a_n), (b_n) are rational sequences, and lim a_n = a, lim b_n = b, show that: lim(a_n+b_n) = a+b, and lim(a_n b_n) = ab. 3..

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