I believe that the example below shows that it is possible that Var(X\_n) goes to infinity and the X\_n converge almost surely.

Suppose that Y is uniformly distributed over the interval \[0,1\] and assume that

P(X\_n = n) =1/2 and P(X\_n= -n)=1/2 if Y < 1/n, and

X\_n = 0 if Y >= 1/n for all positive integers n.

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For any positive integer N,

P( X\_n = 0 for all n>=N) = P( X\_N=0) = P(Y>=1/N) = 1-1/N,

so

P( lim\_n X\_n = 0) >= 1-1/N for all N,

thus the X\_n converge almost surely to 0 and

Var(X\_n) = 1/n\*n\^2= n.

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In the example above, the X\_n are very dependent. I wonder if the following is true or false.

"If Var(X\_n) is not bounded and the X\_n are independent, then the X\_n do not converge almost surely."