I'm trying to show that if a sequence converges it has only one limit point. I can see how to do it by contradiction by assuming that there are 2 limit points and constructing an $\epsilon$.
I was wondering if I could just use the definition of convergence which would make a much more succinct proof.
Let $a_{n} \to a$. Then by definition of convergence, for every $\epsilon > 0$, there exists an integer $N$ such that $|a_{n} - a| < \epsilon$ for every $n \geq N$. Does this imply that $a$ is the only limit point of $\{a_{n}\}$? Since it is true for every $n \geq N$, it must be true that there exists an $n \geq N$ such that $|a_{n} - a| < \epsilon$.