There are infinitely many positive rational numbers. It does not formally make sense to talk about 'half' of them. The way you chose your method, half are lying there. But we can choose other sequences in different ways where it won't happen. I recommend reading up some more on infinities.

Also think about this: using similar argument as you, I can also show more than half of them lie in $[0,\frac{1}{2}]$, then again you can show that for $[0,\frac{1}{4}]$ and so on. Now you can see that will lead to some problems of you keep going like this.

Hope this helps :)

There are infinitely many positive rational numbers. It does not formally make sense to talk about 'half' of them. The way you chose your method, half are lying there. But we can choose other sequences in different ways where it won't happen. I recommend reading up some more on infinities.

Also think about this: using similar argument as you, I can also show more than half of them lie in $[0,\frac{1}{2}]$, then again you can show that for $[0,\frac{1}{4}]$ and so on. Now you can see that will lead to some problems if you keep going like this.

Hope this helps :)