Question

Find ⋃ 𝑨𝒊 ∞ 𝒊=𝟏 and ⋂ 𝑨𝒊 ∞ 𝒊=𝟏 where:

𝑨𝒊 = {𝒊,𝒊 + 𝟏,𝒊 + 𝟐, … } for every positive integer 𝒊.

Answer 1

Let us find $\cup_{i=1}^{\infty}A_i$ and $\cap_{i=1}^{\infty}A_i$ where $A_i = \{i,i+1,i+2, … \}$ for every positive integer $i$.

Taking into account that $A_i\supset A_j$ for $j>i,$ we conclude that $\cup_{i=1}^{\infty}A_i=A_1=\{1,2,3,\ldots\}.$

Let us show that $\cap_{i=1}^{\infty}A_i=\emptyset$ using the method by contradiction. Suppose that $k\in \cap_{i=1}^{\infty}A_i$ for some positive integer $k$. Since $k\notin\{k+1,k+2,\ldots\}=A_{k+1},$ we conclude that $k\notin\cap_{i=1}^{\infty}A_i.$ This contradiction proves that $\cap_{i=1}^{\infty}A_i=\emptyset.$