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1. (10 points) Let L ∈R and let {an }n be a sequence of real numbers such that all its monotone subsequences converge to L. Prove that {an }n converges to L. Is the converse of this statement true (formulate and prove it if so)? 
2. (a) (3 points) Write a mathematical deï¬nition of what it means for a function f : R → R to “be negative for suï¬ƒciently large x”. 
(b) (7 points) Let f : R → R be a continuous function such that f (0) = 1 and such that f (x) is negative for suï¬ƒciently large x. Prove that there exist at least two distinct real numbers a such 
that f (a) = a2 .
3. (a) (8 points) Prove that if f, g : R → R are bounded and uniformly continuous, then the product function f g is also uniformly continuous on R.

Subject  Mathematics 
Due By (Pacific Time)  11/18/2014 12:00 am 
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