Need help on a proof from baby rudin

samspotting

In Theorem 2.41, the one with the heine borel theorem, I do not understand the second inequality,

|x_n - y| >= |x_0 - y| - 1/n >= 1/2*|x_0 - y| for all but a finite n.

why did the 1/n turn |x_0 - y| into 1/2*|x_0 - y|?

I understand this inequality to be using the triangle inequality to show that if the set S has x_0 as a limit point, then by its definition it cannot contain any other limit points, but I don't get how this inequality was made. If this is true for all but a finite n, then it means that once n becomes big enough, then this inequality is true right? When is n big enough?

Why does the inequality not reduce to just |x_0 - y|, as once n becomes big enough 1/n is just 0.

qspeechc

Isn't |x_0 - y| >= 1/2*|x_0 - y| always?

samspotting

Yes, but if the proof was to state that the elements of the set S (that is, x_n for all n) will never get that close to y, then why could he not just eliminate 1/n, since |x_0 - y| is a constant.

Why does the inequality end with 1/2*|x_0 - y|.

My confusion over this leads me to think that I have gotten this proof all wrong.

Office_Shredder

This is a fairly common technique. As n gets large, 1/n becomes small. Eventually, 1/n<= 1/2*|x₀ - y|. When 1/n is smaller than that, you get |x_0 - y| - 1/n >= 1/2*|x_0 - y|.

samspotting

Why not just modify the proof and eliminate 1/n altogether as n approaches infinity.

Why did rudin choose 1/2?

Office_Shredder

1/2 is arbitrary. You can pick anything you want that's less than 1