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# hw1 - a 2 n converges yet a n does not converge(b Let a n =...

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Math 410 - Homework 1 - Due Tuesday June 7 (1) Use the principle of Mathematical Induction to prove that for a natural number n , n X j =1 j 3 = n ( n + 1) 2 2 (2) Follow the steps to prove that 5 is not a rational number: (a) Argue that a rational number can be written m/n such that either m or n is not divisible by 5. (b) Let n be an integer. Show that if 5 divides n 2 then 5 divides n . (c) Follow the proof of proposition 1.2 to show 5 is not rational. (3) Define S = { x | x Q , x > 1 / 2 } . Show that inf S = 1 / 2. (4) Show that | a | - | b | ≤ | a - b | (Hint: Use triangle inequality to get | a | ≤ | a - b | + | b | and repeat with a and b swapped). (5) Use induction to show that ( ab ) n = a n b n . (6) (a) Give and example for which
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Unformatted text preview: { a 2 n } converges, yet { a n } does not converge. (b) Let a n = 2 /n-1 /n 2 + 5. Find the smallest N such that | a n-5 | < 1 / 100 for n ≥ N . (c) Use an ±-N argument to show that a n converges to 5. (7) Show that the interval (1 , 5] is not closed (8) Show that if x n converges to c then so does x n +1 . (9) Let x 1 = 2 and deﬁne x n +1 = 1 6 ( x n + 4). (a) Show that x n +1 < x n is equivalent to x n > 4 / 5. (b) Use induction to show that x n > 4 / 5 and conclude that x n is decreasing. (c) Does x n converge (show why or why not)? If so, ﬁnd the limit....
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