Let $f(n)$ be a converging sequence in $\mathbb{R}$. Now, consider the following inequality involving limit $$ \lim_{n \to \infty} f(n) > a $$
My understanding for the expression above is that there exists an integer $N$ such that for all $n \geq N$, we have $f(n) > a$.
Am I unwrapping the definition correct?
What that inequality means is that the number $\lim_{n\to\infty}f(n)$ is greater than the number $a$. But, in fact, it can be proved that this occurs if and only if, for some number $a'>a$,$$(\exists N\in\mathbb N)(\forall n\in\mathbb N):n\geqslant N\implies f(n)>a'.$$But this is something that has to proved; it is not a trivial consequence of the definition of limit (although it is not hard to prove).