Is there such thing as a "smallest positive number that isn't zero"? My brother and I have been discussing whether it would be possible to have a "smallest positive number" or not and we have concluded that it's impossible. 

Here's our reasoning: firstly, my brother discussed how you can always halve something, $(1, 0.5, 0.25, \dotsc)$. I myself believe that it is impossible because of something I managed to come up with. You can put an infinite amount of zeroes in the decimal place before a number, $(0.1, 0.01, 0.001, \dotsc)$. I am not entirely sure if our reasoning is correct though. I have been told that there is a smallest number possible but I decided to see for myself.
A simple proof by contradiction works here.


Suppose that $a$ is the smallest positive real number.
Next, divide it by $n$ (where $n&gt;1$) to get $\displaystyle\frac a n$.
This new number is smaller than $a$.


Your brother choose $n=2$, while you chose $n=10$.

So we can deny the existence of a smallest positive real number since


  ... there is a "smallest" number and yet there is a number smaller than it.


Same argument works with positive rational numbers.
Your argument and your brother argument both Are right!
Let you assume N is the set of all positive number.

Now let a,b$\in Q$ where Q is a set of all rational number.

Now ,$\frac{a+b}{2}$ is again a rational number .
You can prove it by taking a =$\frac{p}{q}$,b=$\frac {r}{s}$.
And use properties of fraction.

Now we assume a=0,b=1.
Now $\frac{0+1}{2}$ $\in Q$.which is greater than 0 and so it is positive real number.now take $\frac{0+1}{2}$=0.5 as c.
Then $\frac{0+c}{2}$ is again greater than zero and $\in Q$.
Therefore between two rational number there are infinite possibilities to find a new rational number .similarly for irrational number too.THERFORE YOU CAN'T FIND THE SMALLEST POSITIVE NUMBER .but 0 is the greatest lower bound of set of positive real numbers.