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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?

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    $\begingroup$ What you want to achieve is not true. Consider $X_n = n \cdot 1_{(0,1/n)}$ on the unit interval. Regarding the "additional condition": What kind of condition are you looking for? $\endgroup$ – PhoemueX Feb 4 '15 at 17:43
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    $\begingroup$ Also, you should check out Egoroff's theorem (en.wikipedia.org/wiki/…). This is along the lines of what you managed to show. The whole wiki article could be interesting to you. $\endgroup$ – PhoemueX Feb 4 '15 at 17:51
  • $\begingroup$ Ah thank you a lot! I am not completely sure what condition I am looking for. Basically I want to understand the proof of Lemma 3 in the following paper (arxiv.org/pdf/1407.0185v1.pdf), which seemed to apply an argument along these lines, but then I probably misunderstood their argument. $\endgroup$ – air Feb 4 '15 at 17:58

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