Determine an upper bound for the cardinality of $B=\{(a,b,c,d): a,b,c,d \text{ has property } D\}\subseteq [T]^4$, where $[T]=\{1,\dotsc, T\}$.

Question

Say that two natural numbers $a_1$ and $a_2$ has a property $D$ if $\lvert a_1-a_2 \rvert\leq C_T$ for some sequence $C_T$ (here, what matters for the question is that this property is defined for pairs of natural numbers). Define the set $B=\{(a,b,c,d): a,b,c,d \text{ has property } D\}\subseteq [T]^4$, where $[T]=\{1,\dotsc, T\}$. I want to determine an upper bound for the cardinality of $B$.

My strategy

I obtained an upper bound for the cardinality of the set $A=\{(a,b): a,b \text{ has property } D\}\subseteq [T]^2$, i.e., $\#A\leq M$ for some constant $M$. By observing that $B\subseteq \{(s_1,s_2):s_1,s_2\in A\}$, it follows that $\#B\leq M^2$. This inclusion holds since $\{(s_1,s_2):s_1,s_2\in A\}$ does not require that all four numbers have property $D$, pairwise.

Although the upper bound I obtained is exactly the same of others, I would Like to confirm if you agree with this approach.

Thanks in advance.

Answer 1

Since B subset [T]$^4$, an upper bound of |B| is T$^4$.
Since A subset [T]$^2$, an upper bound of |A| is T$^2$.

Exercise. Prove if A subset B, then |B| is an upper bound of |A|.