Edit: Well, this is awkward, after asking the question (that did frustrate me for a few days) I was able to solve it myself without using (and even before looking at) the solutions provided. What am I supposed to do now?
Let $\{a_n\}_{n=1}^\infty$ a sequence converging to $L$. Prove that: c) If $\forall n\in\mathbb{N}$,$a_n\in\mathbb{Z}$ then $L\in\mathbb{Z}$
What I think I should do is: Assume for contradiction that $L\notin\mathbb{Z}$, Therefore exists $a\in\mathbb{Z}$ such that $a-1
$L-\epsilon
Therefore I need to find an epsilon such that:
$a-1\leq L-\epsilon
But I really have no idea what to do here, I can only think of epsilons that are true for one side ($\epsilon=L-a+1$ or $\epsilon=a-L$)