[Math] Show that a certain set of positive real numbers must be finite or countable

Question

Let $B$ be a set of positive real numbers with the property that
adding together any finite subset of elements from $B$ always gives a sum of $2$ or less. Show that $B$ must be finite or at most countable.

$B$ = {$x \in R:x>0\}$, $x_1,x_2…x_n \in B$ such that $x_1+x_2+…+x_n \le 2$.

Question: for any $a,b$ $(a,b)$~$R$, but $B$ is $(0,+\infty)$ so why $B$ is not uncountable (taking as $a = 0$, and letting $b$->$\infty$)?

And why for $B$ being countable doesn't contradict: for any $a,b$ $(a,b)$~$R$?

P.S. I read Showing a set is finite or countable and understood it.

Answer 1

Hint 1: How many elements of $B$ can be in the set $[2,\infty)$?

Hint 2: How many elements of $B$ can be in the set $[1,2)$?

Hint 3: How many elements of $B$ can be in the set $[0.5,1)$?