Generalized Euler's Formula for number of pieces?

Question

I am trying to generalize Euler's formula ($f+v-e=2$) for multiple pieces (pieces meaning different parts with no edges connecting the parts). I decided to do induction on the number of pieces, base case $ \text{number of pieces }= 1$ for which Euler's formula can be used. Then for the induction step I plan on assuming the theorem $f+v-e=k+1$ for $k=\text{number of pieces }$. I wanted to say it works for $\text{number of pieces } < k$ and then try to prove it works for $ \text{number of pieces } = k$. Then I treat each piece as an individual graph and do math (which I don't want to leave to the public internet) to say $f+v-e=2k-k+1$, which is obviously $k+1$. Then I wanted to add an edge, which obviously decreases the number of components, and rebalance the equation. I planned on saying that it is impossible to have a $k=0$ case this way because I acknowledged a $k=1$. Is this a valid proof or did I fall into some invalid induction (induction trap)?