"...all but...", exclusion or inclusion?

Question

Math homework: "... Prove that all but a finite number of the terms of Xn must be...".So I'm not quite sure about what it's asking to prove: prove that "a finite number of the terms..." or all that except the finite number of terms... Thank you.(If complete context is needed for discussion, I will provide it.)

CalifJim · Accepted Answer

all but ~ all except

Prove that all of the terms except 1 or 2 or 3 or ... terms of Xn must be ...
~
Prove that only a finite number of the terms (1, 2, 3, ...) of Xn are exceptions to the idea that all of the terms of Xn must be ...
~
Prove that all of the terms of Xn must be ... except for a finite number of them (1, 2, 3, ...).

You've made this a little difficult to answer by not providing an entire sentence. Who knows what twists and turns await us in the next phrase that might negate everything we've read so far.

CJ

Raen · Answer

Thank you very much CJ. You are right, without the complete sentence, it's difficult to be 100% sure (I definitely feel this way with math analysis most of the time.)

"Let x be in O, where O is an open set. If (x_n) is a sequence converging to x, prove that all but a finite number of the terms of (x_n) must be contained in O."

, but if there's anything you need to add, I'd really love to hear it. Thank you.

CalifJim · Answer

RaenLet x be in O, where O is an open set. If (xn) is a sequence converging to x, prove that all but a finite number of the terms of (xn) must be contained in O.

Yes. It's as I said earlier. There is nothing in this full statement that contradicts the meaning that I told you about earlier.

I think it's saying this:

If the sequence converges to an element in O, then an infinite number of its terms must be in O, though some terms may be outside of O, but the number of them that are outside of O is not infinite. Prove this.

It wouldn't be a bad idea to ask your teacher about this, though, because I'm not so sure about the underlined part. I assume this is topology, and that's not my area of expertise!

CJ

Raen · Answer

Thank you CJ. I'm certain your interpretation is right and it makes sense that the infinite number of terms are in the set (thus causing the sequence to converge because of the property that the set is open). The "..all but..." threw me off, I had thought it meant just the opposite. And yes, this is a topic of topology. Thank you very much for your help.

"...all but...", exclusion or inclusion?

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