hw7hints

hw7hints - Math 240A Real Analysis Fall 2011 Homework...

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Unformatted text preview: Math 240A: Real Analysis, Fall 2011 Homework Assignment 7 Hints 1. Exercise 23 on page 59. Hint. See the hints in the text and also your class notes. 2. Exercise 38 on page 63. Hint. For part (b) only. First, we have f n g n- fg = ( f n- f )( g n- g ) + ( f n- f ) g + f ( g n- g ) . For any σ > 0, one verifies that {| f n g n- fg | ≥ σ } ⊆ braceleftBig | ( f n- f )( g n- g ) | ≥ σ 3 bracerightBig uniondisplay braceleftBig | ( f n- f ) g | ≥ σ 3 bracerightBig uniondisplay braceleftBig | f ( g n- g ) | ≥ σ 3 bracerightBig ⊆ braceleftbigg | f n- f | ≥ radicalbigg σ 3 bracerightbigg uniondisplay braceleftbigg | g n- g | ≥ radicalbigg σ 3 bracerightbigg uniondisplay braceleftBig | ( f n- f ) g | ≥ σ 3 bracerightBig uniondisplay braceleftBig | f ( g n- g ) | ≥ σ 3 bracerightBig . Now show that (replacing σ/ 3 by σ for convenience) μ ( {| f ( g n- g ) | ≥ σ } ) → ....
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This note was uploaded on 12/11/2011 for the course MATH 240a taught by Professor Rothschild,l during the Fall '08 term at UCSD.

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hw7hints - Math 240A Real Analysis Fall 2011 Homework...

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