I am trying to answer the following question:
Prove that $(R \cap S)^{n} \subseteq (R^{n} \cap S^{n})$ for all $n\geq1$
Attempt using Induction
Base Step
$(R \cap S)^{1} \subseteq (R^{1} \cap S^{1})$
$R \cap S \subseteq R \cap S$ (obviously true)
Induction Assumption
Assume that $(R \cap S)^{k} \subseteq (R^{k} \cap S^{k})$ for some arbitrary $k \geq 1$
Prove for k+1
$(R \cap S)^{k+1} \subseteq (R^{k+1} \cap S^{k+1})$
Here, I'm not sure where to use the our assumption that it is true for $n = k$
Attempt Using Mathematical Logic
The relation $R^{n}$ on set A contains all $(a,b)$ such that $a R^{n} b$ for some $a \in A,b \in A$.
Similarly, the relation $S^{n}$ on set A contains all $(c,d)$ such that $c S^{n} d$ for some $c \in A,d \in A$.
$R\cap S$ on set A contains all $(e,f)$ such that $e R f$ and $e S f$ for some $e \in A,f \in A$.
How do I show what $(R\cap S)^{n}$ is, and prove it is a subset of $R^{n} \cap S^{n}$?