Let $(X,M, µ)$ be a measure space, and let $A_j$ be a sequence in $M$ satisfying $µ(A_j∆A_k) ≤ 1/2^j$ for all $1 ≤ j ≤ k$. Show that $∃A ∈ M$ such that $µ(Aj∆A) → 0$ as $j → ∞$.
My attempt:
Let $A := lim inf_{j→∞} A_j := ∪_{n=1}^\inf ∩_{j=n}^\inf A_j$
Now, $A$ is an increasing sequence
$µ(Aj∆A_k) → 0$ as $j,k→ ∞$ With $A$ as defined above, it follows that $(Aj∆A)\subset (Aj∆A_k)$ for $j,k → ∞$
By subadditivity of measure, $µ(Aj∆A)\leq µ(Aj∆A_k)\leq 1/(2^j)$
For $j,k → ∞$, we can find $\epsilon$ such that $µ(Aj∆A_k)\leq \epsilon \implies µ(Aj∆A)\leq \epsilon \implies µ(Aj∆A) → 0$ as $j → ∞$
. I am not sure if I have done it correctly. Please suggest.