Proof that base -2 with binary digits can form every integer

Basically the question is proving that you can create all integers with binary but instead using $-2$ as the base to be able to create negative integers.

Exact question:

Prove that every integer (positive, negative, or zero) can be written as the sum of distinct powers of $−2$.

I somewhat get how you can induct upon increasing powers for $2^0+2^1+2^2$ etc and prove that it will always hold for the next number but I'm not sure how this will work with negative integers since If I induct upwards I can't go down and I can't start at $-\infty$.

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