is this true for an induction proof If I prove $P(0),$ "for all $n: P(n) \Rightarrow P(n+2)$", and "for all $n: P(n) \Rightarrow P(n+3)$", it still may not be true that $P(n)$ is true for all naturals $n.$
If so how can we be sure about $n+1.$

If you prove $P(0)$ and $\forall n(P(n)\to P(n+2))$ then you have proven $P$ is true of any even natural number. (Since we have $P(0)$ and $P(0)\to P(2),$ and thus $P(2),$ and we have $P(2)\to P(4)$ and thus $P(4)$ and so on.) 
Similarly, if you prove $P(0)$ and $\forall n(P(n)\to P(n+3))$ then you have proven $P$ is true of any natural number that is a multiple of $3.$ 
Generalizing, if you prove $P(0)$ and $\forall n(P(n)\to P(n+k))$ then you have proven that $P$ is true of every multiple of $k.$ 
When you do so for $k=1,$ you show it for every multiple of $1,$ i.e. every natural number.
