Stock Option induction problem Can anyone help me solve this problem. I have no idea where to even start on it. Link inside stock option problem


You are asked to prove that
$$
\int_{ - \infty }^\infty  {V_{n - 1} (s + x)dF(x)}  > s - c,
$$
for all $n \geq 1$. For $n=1$, substituting from the definition of $V_0$, you need to show that
$$
\int_{ - \infty }^\infty  {\max \lbrace s + x - c,0\rbrace dF(x)}  > s - c.
$$
For this purpose, first note that 
$$
\max \lbrace s + x - c,0\rbrace  \ge s + x - c.
$$
Then, by linearity of the integral, you can consider the sum
$$
\int_{ - \infty }^\infty  {(s - c)dF(x)}  + \int_{ - \infty }^\infty  {xdF(x)} ,
$$
from which the assertion for $n=1$ follows. 
To complete the inductive proof, substituting from the definition of $V_n$, you need to show that
$$
\int_{ - \infty }^\infty  {\max \bigg\lbrace s + x - c,\int_{ - \infty }^\infty  {V_{n - 1} (s + x + u)dF(u)} \bigg\rbrace dF(x)} > s - c,
$$
under the induction hypothesis that, for any $s \in \mathbb{R}$,
$$
\int_{ - \infty }^\infty  {V_{n - 1} (s + x)dF(x)}  > s - c.
$$ 
For this purpose, recall the end of the proof for $n=1$. 
