How to get comfortable with proofs includes algebra and inequalities

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I'm self studying Baby Rudin and trying to do exercises, i got stuck at exercise 2.18 then looked up the answer from solution manual but the proof includes so much algebra for me to understand where these variables came from and why it is written this way. How can i improve myself for easily seeing the reasoning at this kinds of proofs? This exercise is just an example for kinds of proofs i am having trouble. enter image description here