Prove a thm on stopped processes given fundamental principle 'you can't beat the system'?

Question

How does the principle below imply the thm below?

From Williams' Probability w/ Martingales:

---

Principle:

---

---

Thm:

---

What I tried:

$$E[X_{T \wedge n} - X_0 | \mathscr{F_m}] =/ \le X_{T \wedge m} - X_0 \ \forall m < n$$

$$\to E[X_{T \wedge n} | \mathscr{F_m}] =/ \le X_{T \wedge m}$$

$$\to E[X_{T \wedge n} | \mathscr{F_0}] =/ \le X_{T \wedge 0} \ \text{by choosing m = 0}$$

$$\to E[X_{T \wedge n}] =/ \le X_0$$

$$\to E[E[X_{T \wedge n}]] =/ \le E[X_0]$$

$$\to E[X_{T \wedge n}] =/ \le E[X_0]$$

I think $X_0$ is constant and $\mathscr{F}_0$ is the trivial $\sigma$-algebra.

Is that right?

Answer 1

Hint: Choose the process $C$ so that $(C\cdot X)_n=X_{T\wedge n}-X_0$, which is what you did.

Intuition: To obtain the stopped-at-$T$ process, you want to keep betting so long as time $T$ hasn't yet passed.