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.
For any positive integer N,
P( X\_n = 0 for all n>=N) = P( X\_N=0) = P(Y>=1/N) = 1-1/N,
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.
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."