Answer
See below.
Work Step by Step
1. Let $P(n)$ be the statement to be proved.
2. Test for $n=1,2,3$, we have $d_1=\frac{9}{10},d_2=\frac{10}{11},d_3=\frac{9}{10}\cdot\frac{10}{11}=\frac{9}{11}$ all are within $(0,1]$, thus $ P(1),P(2),P(3)$ are true.
3. Suppose it is true for $n\le p, (p\gt3)$.
4. For $n=p+1$, we have $d_{p+1}=d_p\cdot d_{p-1}\le (1)(1)=1$ and it is within $(0,1]$..
5. Thus $P(p+1)$ will also be true and we have proved the statement using mathematical induction.