Is it valid in an induction to prove a base case for$ n=3$then prove for $n=4$ and use the fact that the transition from $n=3$ to $ 4$ was possible to then prove it is possible for $n=k$ and $ n=k+1?$
For example, If I have the problem to prove that every permutation in $S_n$ is the product of a transposition of the form $(J,J+1)$ where $1\leq J \lt n.$ Can I do this.... Base case, $n=2$ works (show work) $n=3$ works (show work) inductively assume n=k works and say because the transition from $n=3$ to $ n=4$ worked then $n=k$ to $n=k+1$ will work?
Not valid. This would fail for example if we have 1,1,2,3,5,8,... and the statement was the nth number is odd. Then n=3 is odd, n=4 is odd, but the rest of the numbers are not odd.