I have to prove the following:
"For every even integer $0\leq a\leq2^n-2$ (and for all integers $n\geq1$), there exist binary ($0$ or $1$) $w_i$ such that $\sum_{i=1}^{n-1} (w_i2^i)=a$"
I plan to prove this statement by writing out the numbers $0,1,2,3,4,5,6,7,...$ in binary format
00000
00010
00100
00110
01000
01010
01110
10000
...
and noting that if we consider, for example, each digit of the binary number $00110$ (corresponding to 'base $10$' number $3$) as the value of $w_i$ (where $i = 1$ corresponds to the digit immediately before the rightmost digit of the binary number) when $n=3$, we associate each binary number with an even integer $\sum_{i=1}^{n-1} (w_i2^i)=a$.
I want to show that the correspondence between the binary numbers and numbers $0,1,2,3,4,5,6,...$ implies that we can associate a binary number (representing the $w_i$) with each even integer in the desired range (for all $n$ in the stated range).
Is this a valid approach to prove the statement in quotes above? If not, how can I start the proof? Would I need to use induction and/or contradictions?
Any feedback would be greatly appreciated.