A metric space in which $3^\infty=2^\infty=0$

Question

I want a space containing all the positive integers in which $3^nx+3^n-2^n\to0$ as $n\to\infty$

Perhaps paradoxically, numbers not factorisable by $2,3$ would be a sufficient set for me (in case that helps).

My rudimentary knowledge says that a sum of two metrics is a metric and therefore I can just take $d(x,y)=\lvert x-y\rvert_2+\lvert x-y\rvert_3$

Am I going about that the right way?

Is this space going to have reasonable properties?

Answer 1

You can define a metric $d$ on $[0,\infty)$ by \begin{align*} d(x,y) &= \left| \frac 1x - \frac 1y \right|,\quad x,y > 0 \\ d(x,0) = d(0,x) &= \frac 1x ,\quad x > 0 \\ d(0,0) &= 0.\end{align*}

Then $d(3^n x + 3^n - 2^n,0) \to 0$.