Proof by induction. Let $f(0)=1$ and $f(n+1)= (n+1)f(n)$. Prove $f(n)=n(n-1)(n-2)...(2)(1)$

Question

It says Proof by Induction, Strong induction or Well-order but I don't know how can i start.

I tried proving for $f(1)= f(0+1)= 1(f(0))=1\times 1$ but $1(1-1)=0$ then I don't know what can I do

Answer 1

$\text{Base Case: }n=0$ $$f(1+0)=(1+0)f(0)=f(0)=1$$

You can also, for your own edification, do it for $n=1$ (recalling the result from above):

$$f(1+1)=f(2)=(1+1)f(1)=2\times 1=2$$

From here, you can do the induction step:

$\text{Induction Step: Given that this holds for $(n-1)$, prove it holds for $n$}$, or:

$$f(n-1+1)=(n-1+1)f(n-2)=n!\rightarrow f(n+1)=(n+1)!$$

Can you take it from here?