[Solved]: What is the difference between “for infinitely many n” and “for all n”?

“for infinitely many input sizes $n$, $P(n)$ holds” means there are input sizes $n_1$, $n_2$, $ldots$, (infinitely many) such that $P(n_i)$ holds for all $i$. In other words, for every integer $k$, there is an input length $n geq k$ such that $P(n)$ holds. In contrast, “for all input sizes, $P(n)$ holds” means that for every input size $n$, $P(n)$ holds. Typically, “for all input sizes, $P$ holds” and “for almost every input size, $P$ holds” are used interchangeably, although the latter technically means that there is a $k$ such that for all input sizes $n geq k$, $P$ holds. (Note that the negation of “for infinitely many input sizes, $P$ holds” is “for almost every input size, $neg P$ holds.”) This is generally because in the models of computation being considered, constant factors don’t matter, so from an “almost every input size” statement you can generally hardcode the solutions for all inputs of length less than $k$ in your model, and get a “for all input sizes” statement.