hw5 - then A B R n + p is compact . 7. Let U R n be open,...

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Homework 5 – Math CS 120, Winter 2009 Due on Thursday, February 12th, 2009 1. Consider R n , and the norms k · k p , for 1 p < . (a) Find constants c p > 0 , d p > 0 such that c p k x k p ≤ k x k d p k x k p , x R n . (b) For any 1 p, q ≤ ∞ , find constants c p,q > 0 , d p,q > 0 such that c p,q k x k p ≤ k x k q d p,q k x k p , x R n . 2. Prove that a sequence x k = ( x 1 ,k | , x 2 ,k , . . . x k,n ) in R n , k R converges to z = ( z 1 , . . . , z n ) R n in the norm k · k p if and only if { x l,k } k N R converges to z l in R , i.e., if and only if every component converges. 3. Prove the Heine-Borel theorem: If A R n is a compact set and f : A R m is continuous, then f is uniformly continuous . 4. Prove the following theorem: If A R n is a compact set and f : A R m is continuous, then f ( A ) is compact . 5. Prove the following theorem: If A R n is a compact set and f : A R is continuous, then f achieves its maximum and minimum in A , i.e., there exist z 0 , z 1 A such that f ( z 0 ) = min { f ( x ) | x A } , f ( z 1 ) = max { f ( x ) | x A } . 6. Prove the following theorem: If A R n and B R p are compact sets,
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Unformatted text preview: then A B R n + p is compact . 7. Let U R n be open, and consider a function f : U R m . Prove that f is continuous if and only if A R m open set, f-1 ( A ) R n is open. 8. Determine whether the following series converges or diverges: X n =2 1 n (log n ) 2 . 1 2 9. Prove that if lim n x n = x, then lim 1-(1- ) X n =1 x n n = x. Prove that the converse is, in general, false by computing the second limit for x n = (-1) n . 10. Consider the series of real numbers X n =1 a n . Prove that if lim sup n a n +1 a n = r &lt; 1 , the series converges, and that if r &gt; 1, the series diverges. Find two examples, one converging and one diverging, for which r = 1....
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hw5 - then A B R n + p is compact . 7. Let U R n be open,...

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