4. (12 points) Let a ER be an irrational number, and let S = {m+na| m,...

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4. (12 points) Let a ER be an irrational number, and let S = {m+na| m, n = Z}. (a) Prove that ra + sb € S for all a, b E S and all r, s € Z. (b) Prove that there exists a strictly monotone increasing sequence of natural numbers {n} such that the sequence {na - [na]}=1 k=1 converges. (c) Prove that for every e > 0, there exists an element 7 € S such that 0 < T < E. (d) Prove that S is dense in R; that is, for every x ER and every > 0, there exists a point y ES such that x - y < e. Hint. [x/T] ≤x/T ≤ (x/T] +1. 4. (12 points) Let a ER be an irrational number, and let S = {m+na| m, n = Z}. (a) Prove that ra + sb € S for all a, b E S and all r, s € Z. (b) Prove that there exists a strictly monotone increasing sequence of natural numbers {n} such that the sequence {na - [na]}=1 k=1 converges. (c) Prove that for every e > 0, there exists an element 7 € S such that 0 < T < E. (d) Prove that S is dense in R; that is, for every x ER and every > 0, there exists a point y ES such that x - y < e. Hint. [x/T] ≤x/T ≤ (x/T] +1.

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a Since a b S a mna and b mnb for some m n Z Then rasb rmsnasnb r... View the full answer