Difference between for all but a finitely number of n and infinitely number of n

So for the statement P(n)P(n) to occur for all but a finitely number of nn, there should be a number kk such that after n>kn>k P(n)P(n) occurs.

What I didn’t quite get is that, after n>kn>k there are infinite number of nn. So P(n) also occurs for infinitely many nn?

Can someone clearly explain me the difference between them

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“all but a finitely number” means that there is an NN such that P(x)P(x) is true for all nn greater than NN. From that number on there are no more numbers such that PP does not hold for them.
– Mauro ALLEGRANZA
Oct 20 at 18:22

  

 

On natural numbers, to occur infinitely oftem may happen three different ways. i) To always occur; i.e. to never not occur. ii) to occur infinitely but to not occur a finite number of times. iii) to occur infinitely but also not occur infinitely. To occur all but finitely, does indeed mean it occurs infinitely, but to occur infinitely doesn’t mean it only fails to occur finitely.
– fleablood
Oct 20 at 18:48

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3 Answers
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Anything true for all but finitely many nn is true for infinitely many nn (for the reason you mention).

Consider the statement: “nn is odd”. It’s true for infinitely many nn, but not true for all but finitely many nn.

Saying that P(n)P(n) is true for all but finitely many n∈Nn\in\mathbb{N} is a stronger statement than saying that P(n)P(n) is true for infinitely many n∈N.n\in\mathbb{N}.

If you are a mammal, then you are an animal. If you are an animal, you may or may not be a mammal.

If P(n) always happens or fails to happen only a finite number of times then, yes, P(n) happens an infinite number of times. But if P(n) happens an infinite number of times, but also fails an infinite number of times, then well, P(n) happens an infinite number of time and fails an infinite number of times.

Emotionally:

P(n) always happens. Hurray! That’s strong and fantastic!!! Yeah, Always! Yeah. Good results.

P(n) fails when n=7, but happens everywhere else. Hurray! 7 is is fluke. This is a dead cinch if we just watch out for n=7.

P(n) fails when M/nM/n is prime, but happens everywhere else. Oh, … well, M/nM/n is prime… how bad is that; how often does it happen? (Well, eventually n>Mn > M and M/nM/n can’t be an integer much less prime; so it only happens a finite number of times.) So, it might fail a lot of times but it passes an infinite amount of time? In the long run it doesn’t matter because eventually it will always happen, right? (Yes) … okay…. Hurray! It happens all but finitely times!

P(n) fails when n=2p−1n = 2^p – 1 and pp prime. Oh… that’s kind of bad, isn’t it? (Not really, n=2p−1n = 2^p – 1 is a pretty unusual.) So, eventually we’ll get past it? And it will always be true. (No, there’ll always be a larger number where it won’t be true.) But they’re rare? Right? (Yes.) … okay… hurray???

P(n) happens whenever nn is even and fails whenever nn is odd. … (What’s wrong? It happens an infinite number of times. Cheer up.) …Yeah, but it fails as commonly as it passes… (Yes, but it doesn’t always fail) … meh… (Say “hurray”) …. hurray *sigh*… whoop-de-effin-do…