# Is this proof correct?

## Homework Statement

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

(a) Write a proof by mathematical induction that $$A_1\subseteq A_n$$ for all n.

(b) Use part a to prove that $$\bigcap$$ from n=1 to infinity of $$A_n = A_1$$

## The Attempt at a Solution

(i) $$A_1\subseteq A_1$$ by some theorem in my book. Any set is a subset of itself.
(ii) Assume $$A_1\subseteq A_n$$ for all n >= 1
Then we know that$$A_n\subseteq A_{n+1}$$ by the given description of the family of sets.
Then $$A_1\subseteq A_n$$ is true by inductive hypothesis, therefore $$A_1\subseteq A_{n+1}$$ 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 $$A_1\subseteq A_{n+1}$$, 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.