Let $(\Omega, \mathscr{F}, \mathbb P)$ be a probability space and $(X_n)_{n \in \mathbb N}$ a sequence of random variables on that space, which converge almost surely to a constant $c\in \mathbb R$, i.e.
$$X_n \, \xrightarrow{\mathrm{a.s.}} \, c$$
I am now interested in whether the following statement holds:
Let $\varepsilon > 0$. Then there exists $N \in \mathbb N$, so that the following holds almost surely:
$$ |X_n - c| \leq \varepsilon \; \forall n \geq N$$
Trying to show this, the closest I got was:
$$ \mathbb P[\omega \in \Omega: |X_m(\omega) -c| \leq \varepsilon \; \forall m\geq n] \to 0, n\to\infty$$
Also if the above statement is not true (which I actually think is the case), can we impose any additional conditions so that it holds?