Let $K$ be a field. Given a map $f\colon K\longrightarrow K$, and $h\not=0$ define $\Delta_h f$ to be the map $x\longmapsto\dfrac{f(x+h)-f(x)}{h}$. Then $\Delta_h^j f$ is defined for $j=0,1,2,\dots$.
Given a sequence $(g_0,g_1,g_2,\dots)$ let $\Delta g$ be the sequence $(g_1-g_0,g_2-g_1,g_3-g_2,\dots)$. Again $\Delta^jg$ is defined for all non-negative integers $j$. Fixing $x\in K$ let $g$ be the sequence such that $g_n=f(x+n\cdot h)$.
I want to show by induction, that $h^j\cdot(\Delta_h^jf)(x)=(\Delta^jg)_0$, but I am getting lost in calculations and I cannot do the induction step. Can you help me?
I suggest to show the general case $h^j \Delta^j_h f(x+kh)=(\Delta^j g)_k$.
Step 1: Show that it is true for $j=0, k=0$;
Step 2: Suppose that it is true for $j=0$, and some $k$, show that it is true for $j=0$, and $k+1$. This shows that it is true for $j=0$ and all $k$;
Step 3: Suppose that it is true for some $j$ and all $k$, show that it is true for $j+1$ and all $k$.