Question

2.D. Let $(A_n)$ be a sequence of subsets of a set $X$. If $A$ consists of all $x \in X$ which belong to infinitely many of the sets $A_n$, show that $\mathcal{A} = \bigcap_{m=1}^{\infty} \left[ \bigcup_{n=m}^{\infty} A_n \right]$. The set $A$ is often called the limit superior of the sets $(A_n)$ and denoted by $\lim \sup A_n$.

          2.D. Let $(A_n)$ be a sequence of subsets of a set $X$. If $A$ consists of all $x \in X$ which belong to infinitely many of the sets $A_n$, show that
$\mathcal{A} = \bigcap_{m=1}^{\infty} \left[ \bigcup_{n=m}^{\infty} A_n \right]$.
The set $A$ is often called the limit superior of the sets $(A_n)$ and denoted by $\lim \sup A_n$.

2.D. Let (An) be a sequence of subsets of a set X. If A consists of all x ∈ X which belong to infinitely many of the sets An, show that
𝒜 = ⋂m=1^∞[ ⋃n=m^∞ An ].
The set A is often called the limit superior of the sets (An) and denoted by limsup An.

Added by Patricia M.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Suman K

06/16/2022

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2.D. Let (A,) be a sequence of subsets of a set X.If A consists of all x e X which belong to infinitely many of the sets An, show that A=n UA The set A is often called the limit superior of the sets (A,) and denoted by lim sup An