What is the difference between "infinitely many $n$" and "large enough $n$"?

I'm faced with this sentence in a solution of a problem:

The negation of "$P(n)$ for infinitely many $n$" is "$\lnot p(n)$ for large enough $n$".

What is the difference between infinitely many and large enough $n$?

It is true that infinitely many integers are even.

But it is not true that all large enough integers are even, because no matter how big an integer you pick, I can always find an odd integer which is bigger.

Perhaps these definitions will clarify the issue:

• We say that $P(n)$ holds for infinitely many $n$ if for any $N$ there is $n>N$ so that $P(n)$ is true.¹
• We say that $P(n)$ holds for large enough (or sufficiently large) $n$ if there is $N$ so that $P(n)$ is true for all $n>N$.

With these definitions, can you see why "$P(n)$ for infinitely many $n$" and "$\lnot P(n)$ for large enough $n$" are negations of each other?

It might be that the source you are using has different definitions. If that is the case, it is best to include the definitions in your question to make sure every speaks exactly the same language. The definitions I gave above are not the only possible way to define these concepts.

¹ Mind you, this is true if we are working over $\mathbb N$, but not in general. For something less context-dependent, you could say that the set $\{n\in\mathbb N;P(n)\}$ is infinite. The reason I chose not to go this way is to make the comparison clearer.

When something is true for infinitely many $n$, it doesn't have to be true always even when $n$ gets massive. For example, $P(n)$ might be "$n$ is a prime". This statement is true for infinitely many $n$, but it's not the case that as soon as $n>n_0$ for some number $n_0$, every $n$ is an prime number.

On the other hand, if something is true when $n$ is large enough, then it's true for infinitely many $n$. This is because there are infinitely many numbers.