<s>[INST] Why are sums of powers of 2 able to give all numbers? It is known that 

If we sum up a combination of numbers that are positive powers of 2(starting from 0 to infinity), we can get any number possible.

(Correct me if this is wrong).
Can anyone give a proof and explain how this works? Please try to make it understandable.
 [/INST] Here's an example, which can be easily expanded to the general case. Let's say we want to write $21$ as the sum of distinct powers of two. ("Distinct" meaning that they're all different.)
Well, if we drop the distinctness condition, it's easy:
$21=1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1$
Let's try to clean this up a bit. We don't need more than one $1$, because we can do this:
$21=1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1\\
\phantom{21}=(1+1)+(1+1)+(1+1)+(1+1)+(1+1)+(1+1)+(1+1)+(1+1)+(1+1)+(1+1)+1\\
\phantom{21}=2+2+2+2+2+2+2+2+2+2+1$
Similarly, we don't need more than one $2$, because we can do this:
$21=2+2+2+2+2+2+2+2+2+2+1\\
\phantom{21}=(2+2)+(2+2)+(2+2)+(2+2)+(2+2)+1\\
\phantom{21}=4+4+4+4+4+1$
And we don't need more than one $4$:
$21=4+4+4+4+4+1\\
\phantom{21}=(4+4)+(4+4)+4+1\\
\phantom{21}=8+8+4+1$
And finally:
$21=8+8+4+1\\
\phantom{21}=(8+8)+4+1\\
\phantom{21}=16+4+1$
Basically, we start with a bunch of ones, and then "collapse" them until all powers of two are different from each other.
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