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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.
Bob deposits quarters and dollar bills into a vending machine to buy snacks. Find a recurrence relation for the number of ways he can deposit 25*n cents into the machine if the order matters.
Kathy bicycles 8 km/h faster than Javier. In the same time, it takes Javier to bicycle 45km, Kathy can bicycle 69km. How fast does each bicyclist travel?
A baseball diamond has the shape of a square with sides of 90ft long. A player 30ft from third base is sunning at a speed 28 feet/sec. At what rate is the player's distance from home plate changing.
Let S be the circle in the plane {(x,y) in R^2: x^2 + y^2 = 1} and f: S --> R be a continuous map. Show that there exists (x,y) on S such that f(x,y) = f(-x,-y).
If the route is randomly selected what is the probability dat she will visit the cities in alphabetical order?
A self-employed worker earned $47,000 during the year. The FICA tax for Social Security is 6.2% and the FICA tax for Medicare is 1.45%.
the runners are running on circular tracks of which has a circumference of 1320 feet. the tracks are 100 feet apart and and the runners start opposite each other and move at the same constant rate of 880 ft/min. how fast are the runners separating..
A farmer goes to the market and buys one hundred animals at a total cost of $1000. If cows cost $50 each, sheep cost $10 each, and rabbits cost 50 cents each, how many of each kind does he buy?
find the distance between two ships from the word problem.from a lighthouse the lighthouse keeper sees two ships.nbsp
If the sum of the areas of both gardens is 165 square yards, find the measure of a side of the square garden.
Saunders Corp. has current liabilities of $440,000, a quick ratio of 0.81, inventory turnover of 5.7, and a current ratio of 1.8. What is the cost of goods sold for the company?
The theoretical probability of undesirable side effects resulting from taking Grebex is 1 in 12. If 600 people take Grebex to lower their blood pressure, how many will encounter undesirable side effects?
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