EDB — 065

Exercises

1. [065]Prerequisites:[064]↺↻, [01G]↺↻. Given sets \(A_ 1,A_ 2\ldots \) and \(B_ 1,B_ 2\ldots \) , for \(n∈ℕ\), say if there is a relation (of equality or containment) between

   \begin{eqnarray} \label{eq:liminf_ insiemi_ intersezione} (\liminf _{n→∞} A_ n) ∩ (\liminf _{n→∞} B_ n) & \stackrel{?}{=} & \liminf _{n→∞} (A_ n∩ B_ n)\quad ,\\ {} \label{eq:liminf_ insiemi_ unione} (\liminf _{n→∞} A_ n) ∪ (\liminf _{n→∞} B_ n) & \stackrel{?}{=} & \liminf _{n→∞} (A_ n∪ B_ n)\quad . \end{eqnarray}

   If equality does not hold, show an example. Then use [063]↺↻ to establish similar rules for \(\limsup _{n→∞} A_ n\).

   Solution 1

   [066]↺↻