- #1

- 128

- 0

## Homework Statement

Let [tex]\left\{A_n | n \in N\right\}[/tex] be a family of sets satisfying [tex]A_n \subseteq A_{n+1}[/tex] for all n >= 1.

(a) Write a proof by mathematical induction that [tex]A_1\subseteq A_n[/tex] for all n.

(b) Use part a to prove that [tex]\bigcap[/tex] from n=1 to infinity of [tex]A_n = A_1[/tex]

## The Attempt at a Solution

(i) [tex] A_1\subseteq A_1[/tex] by some theorem in my book. Any set is a subset of itself.

(ii) Assume [tex]A_1\subseteq A_n[/tex] for all n >= 1

Then we know that[tex]A_n\subseteq A_{n+1}[/tex] by the given description of the family of sets.

Then [tex]A_1\subseteq A_n[/tex] is true by inductive hypothesis, therefore [tex]A_1\subseteq A_{n+1}[/tex] for all n>= 1 by induction.

For part b:

I think it seems very obvious but I'm kind of burned out from working the first one. So I have so far just written down that since [tex]A_1\subseteq A_{n+1}[/tex], then the family of sets from n=1 to infinity include A_1, thus the intersection from said limits of A_n = A_1

But I'm sure there must be some formalism I'm not catching.