# Induction and Union of Sets

I'm trying to prove the following:

" Suppose that one has proven the proposition that if $$A \subseteq B$$ and $$C \subseteq D$$, then $$A \cup C \subseteq B \subseteq D$$. Prove that for any integer $$n \geq 2$$ that if sets $$A_1, A_2,...,A_n$$ and $$B_1, B_2,...B_n$$ are sets that satisfy $$A_j \subseteq B_j$$ for $$j = 1, 2, ..., n$$ then $$\bigcup_{j=1}^n A_j\subseteq \bigcup_{j=1}^n B_j."$$

I'm not sure if I what I came up with makes sense logically, and would appreciate some feedback.

Proof:

Define P(n): $$\bigcup_{j=1}^n A_j\subseteq \bigcup_{j=1}^n B_j$$.

Base case $$(n=2)$$:

$$\bigcup_{j=1}^2 A_j \subseteq \bigcup_{j=1}^2 B_j = A_1 \bigcup A_2 \subseteq B_1 \bigcup B_2$$. So $$P(2)$$ holds.

Inductive step: Assume $$P(k)$$ holds for some $$k \geq 2$$. So $$\bigcup_{j=1}^k A_j \subseteq \bigcup_{j=1}^k B_j = A_1 \bigcup A_2 \bigcup ... \bigcup A_k \subseteq B_1 \bigcup B_2 \bigcup ...\bigcup B_k.$$ Notice, $$\bigcup_{j=1}^{k+1} A_j \subseteq \bigcup_{j=1}^{k+1} B_j = A_1 \bigcup A_2 \bigcup ... \bigcup A_k \bigcup A_{k+1} \subseteq B_1 \bigcup B_2 \bigcup ... \bigcup B_k \bigcup B_{k+1}.$$ So $$P(k+1)$$ holds and by induction $$P(n)$$ holds for all $$n \geq 2$$.

• Please do not rely on images to convey key information about your question. See here for an explanation of why it is frowned upon. Commented Oct 18, 2021 at 1:39
• @ArturoMagidin Oh, I didn't realize this was an issue! Should I edit it, then? Commented Oct 18, 2021 at 1:40
• Yes. Please replace the image with text; you can indent it (using >) to indicate you are quoting. Commented Oct 18, 2021 at 1:41
• You have written out what the claims involve, but you haven’t really shown them to be true. For example, for the base case, you need to show that if $A_1 \subseteq B_1$, and $A_2 \subseteq B_2$, then $A_1\cup A_2 \subseteq B_1\cup B_2$ Commented Oct 18, 2021 at 1:41
• @ArturoMagidin Ok, thank you. Commented Oct 18, 2021 at 1:41

Inductive step: Assume $$P(k)$$, then we want to show it holds for the inductive step $$P(k+1)$$:
$$\bigcup_{j=1}^{k+1} A_j \subseteq \bigcup_{j=1}^{k+1} B_j = \left(A_1 \bigcup A_2 \bigcup ... \bigcup A_k\right) \bigcup A_{k+1} \subseteq \left( B_1 \bigcup B_2 \bigcup ... \bigcup B_k\right) \bigcup B_{k+1}.$$