real analysis, math homework help

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errz2015

Mathematics

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Hello,  this is the hint from my professor, he want me do it by this way.

please can you solve it again by professor's hint

thank you.

   #5 page 264 , #5 page 267

the book methods of real analysis, goldberg and the link the book below:

https://alansinyal.files.wordpress.com/2012/08/method-of-real-analysis.pdf


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Hint # 5 page 264 Hint #5 page 267
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Attached.

Question #5 page 264:
We write





f n ( x ) − f m ( x ) =  f n ( x ) − f m ( x ) −  f n ( x0 ) − f m ( x0 ) +  f n ( x0 ) − f m ( x0 ) , for all x  a, b (*)
Applying 7.7A for g ( x ) = f n ( x ) − f m ( x ) , we see that there exists c lies between x and x0 satisfying

g ( x ) − g ( x0 ) = g  ( c )( x − x0 )
 f n ( x ) − f m ( x )  −  f n ( x0 ) − f m ( x0 ) 

= f n ( c ) − f m ( c )
x − x0





  f n ( x ) − f m ( x )  −  f n ( x0 ) − f m ( x0 )  =  f n ( c ) − f m ( c )  ( x − x0 )


Hence, we can replace the expression  f n ( x ) − f m (...


Anonymous
This is great! Exactly what I wanted.

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