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I am self teaching probability, and would like to follow up on this thread: Distribution of infinite sum of Bernoulli

I understand that the distribution of $S_n$ is a uniform distribution. I'm now working on:

Show that the sequence $S_n$ converges almost surely as n → ∞ [Hint: Cauchy sequences converge]. What is the distribution of the limit S?

I believe that the sequence $S_n$ converges almost surely to 1? However, I struggle with proving almost sure convergence, so please can someone provide a proof? Thanks a lot.

yw_2003
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  • You have not said what $S_n$ is and the linked question talks about $Y_n$ and its distribution converging to a continuous uniform distribution on $[0,1]$. – Henry Oct 18 '22 at 11:09

  • I believe that the sequence $S_n$ converges almost surely to 1?

    If its distribution tends to a uniform, then that cannot be true.

     – leonbloy Oct 18 '22 at 15:03

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