I am having a really hard time grasping proof by induction and struggling to write consitent thorough proofs which use induction.
For example, proving the following
$k! \geq 3^{k-2}$
Now I understand that we first test the base case $k=1$, and see that it holds and then assume that it is true for any $k$, and if we can show it holds for $k+1$, then we would have proven it for all $k>1$.
However, I am really struggling with the inductive step, someone suggested multiplying both sides by $k+1$ leaving $(k+1)! \geq 3^{k-2}(k+1)$ but I still don't see how it helps, surely we should be putting in $k+1$ instead of $k$?
If anyone has any texts I can read or articles I can read to help improve my skills with proof by induction I would really appreciate it, I'm really struggling with it.
You can see that that the inequality holds for $k=1$. Now lets assume it holds true for some $k>1$.Thus we have $$k!>3^{k-2}$$ multiply both sides by $(k+1)$, $$(k+1)!>=3^{k-2}.(k+1)$$ we know $k>=2$ thus $k+1>=3$, so $3^{k-2}.(k+1)>=3^{k-2}.3=3^{k-1}$ so we get $(k+1)!>=3^{k-1}$.Thus relation holds true for $k+1$ also. As far as books are concerned you can look up the chapter about induction (chapter 2) in the book A Walk Through Combinatorics by Bona. It has examples plus a lot of solved exercises.