Show that the set is not countable

Question

To show that a set is countable, you need to show 1 to 1 correspondence, right? So to test if it is 1 to 1 and also onto. So for this example:

R is the set of real numbers. Let S = { x∈R | -3 < x < 0 }.
Show that the set S is not countable.

I can see this set is infinitely big. Also, all I see is that it IS 1 to 1 and onto because for example; 1 -> -2.999, 2 -> -2.990, 3 -> -2.900, ... Obviously, there is an infinite amount of numbers in between each of X values I selected. So you would shift all the numbers.

Z+ goes to infinite and the set goes from -3 to 0, infinitely.

But it says show that the set is not countable, so I am doing something wrong.

Answer 1

Hint : can You find a bijection between $\mathbb{R}$ and $S$?

Answer 2

We have that: $S=\{s\in \Bbb R: -3< s<0\}$

We can also see that: $S= \big\{ \boxed{?} \mathop{\big|} t \in\Bbb R, 0 < t < \infty\big\}$

So there is a bijection between $S$ and the positive reals, and further the positive reals are known to be uncountably infinite.

What goes in the box?