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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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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 (...
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