# hw4 - > 0 such that c p,q k x k p ≤ k x k q ≤ d p,q k...

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Homework 4 – Math 118A, Fall 2009 Due on Thursday, October 29th, 2009 1. Prove that in a metric space ( X, d ), a sequence { p n } n N converges to p X , if and only if every subsequence of { p n } n N converges to p . 2. Find the upper and lower limits of the sequence { s n } de±ned by s 1 = 0; s 2 m = s 2 m - 1 2 ; s 2 m +1 = 1 2 + s 2 m . 3. Prove that for any x , y R n , < x , y > = 1 4 ( k x + y k 2 - k x - y k 2 ) . 4. In R n we de±ned earlier k x k p = n X i =1 | x i | p ! 1 /p , and saw that k·k p is a norm if p 1. Prove that k·k p does not derive from an inner product, unless p = 2, that is to say, there does not exist an inner product in R n such that k x k p = < x , x > 1 / 2 x R n , unless p = 2. 5. Given p 1, ±nd 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 . 6. For every 1 p, q ≤ ∞ , ±nd constants c p,q > 0 , d p,q

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Unformatted text preview: > 0 such that c p,q k x k p ≤ k x k q ≤ d p,q k x k p , ∀ x ∈ R n . 7. Let ( X, d ) be a metric space, and A ⊂ X . De±ne the distance from x to A by d ( x , A ) = inf ± d ( x , y ) ² ² ² ² y ∈ A ³ . Prove that ² ² ² ² d ( x , A )-d ( y , A ) ² ² ² ² ≤ d ( x , y ) ∀ x , y ∈ X. 1 2 8. Let 0 < x 1 < y 1 . For n ≥ 1, set x n +1 = √ x n y n , y n +1 = 1 2 ( x n + y n ) . Show that lim x n and lim y n exist and are equal. ( Hint: Check that < x n < y n , y n +1 < y n , and x n < x n +1 ) ....
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hw4 - > 0 such that c p,q k x k p ≤ k x k q ≤ d p,q k...

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