I'm going crazy trying to understand this problem, like it's literally driving me mad. Please someone help explain this to me?

Use the definition of the limit to show that

So I said "Given any , there exists a such that if , then

So I simplified to , and I get stuck.

The biggest issue I have is that the "answer" (but it really doesn't give you a proof) is in the back of the book, and I'm pulling my hair out trying to understand it.

This is what they say:

If , then

Q: How did they say that . Why were they able to just say that it is less than 1/4?

Then this is where I really get confused:

Q: Where did they get this from? I understand that it is less than that, but I don't get how they got that. Like if I were doing a different problem, would I just pull that out of thin air?

Then they say we may take

Q: I kind of understand this, so that it will always be less than , but I'm again not sure where they got this?

If anyone could help me, that would be awesome. I really want so badly to understand how to do this, I just can't do it alone!

Thank you.