Convergence of a non-increasing sequence of random variables to a constant.

I have a non-increasing sequence of random variables $\{Y_n\}$ which is bounded below by a constant $c$ $\forall \omega \in \Omega$. i.e $\forall \omega \in \Omega$, $Y_n \geq c$, $\forall n$. Is it true that the sequence will converge to $c$ almost surely?

Thanks

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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. – user53153 Jan 1 '13 at 5:48
On the mathematical side, it may be helpful to consider a simpler problem first: suppose we have a non-increasing sequence of real numbers which is bounded below by $c$. Will it necessarily converge to $c$? – user53153 Jan 1 '13 at 5:49

No, of course not. For example, add any positive random variable to all the $Y_n$ and the hypotheses are true (with the same $c$), but the conclusion is not.
Even after adding any positive random variable to all the $Y_n$, if the non-increasing nature is still preserved, why shouldn't it converge to $c$? What I am missing here! – user54772 Jan 1 '13 at 5:47
If $Y_n \to c$, then $Y_n + X \to c + X$, not to $c$. – Robert Israel Jan 1 '13 at 5:53