A problem in symmetric difference from Hewitt and Stromberg

Question

Let $\{M_n:n\in \mathbb{N}\}$ and $\{N_n:n\in \mathbb{N}\}$ be two collection of non-empty sets such that the collection $\{N_n:n\in \mathbb{N}\}$ be pairwise disjoint. Let $Q_n=M_{n} - (\bigcup_{1\leq k\leq n-1} M_k)$. I need to show that $$N_n \triangle Q_n\subseteq \bigcup_{1\leq k\leq n}M_k\triangle N_k$$ for all $n$. Please help.

Answer 1

I'm pretty sure you meant to define $Q_{n}=M_n\setminus \bigcup_{k

Assume first $x\in N_n\setminus Q_n$. Now, since $x\not\in Q_n$, either it is not in $M_n$, in which case $x\in M_n\bigtriangleup N_n$. Or $x\in M_n$, but then $x\in M_k$ for some $k

Otherwise $x\in Q_n\setminus N_n$. But then $x\in M_n$, so $x\in M_n\bigtriangleup N_n$ and thus again $x\in\bigcup_{k\leq n}M_k\bigtriangleup N_k$.

As $x$ was arbitrary, the claim is proved.