real analysis, math homework help

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