The provided image contains an application of Markov's inequality. The main task is to demonstrate that the statement in equation (11) holds for any positive $\varepsilon > 0$ and $\delta > 0$. Let's break down the process and the steps involved:

1. Step 1: Understanding Markov's Inequality
   Markov’s inequality states that for a non-negative random variable $X$, and any $t > 0$: $P(X \geq t) \leq \frac{E[X]}{t}$ In your case, the inequality applies to the expression $R(X_n) - R^*$.

2. Step 2: Breaking Down the Expression
   The inequality given in the image is: $P\left\{ R(X_n) - R^* > \varepsilon \right\} \leq \frac{1}{\varepsilon} E\left[ R(X_n) - R^* \right]$ This is a direct application of Markov's inequality to the random variable $R(X_n) - R^*$ with threshold $\varepsilon$.

3. Step 3: Substituting Expectation
   The expectation is further evaluated as: $E\left[ R(X_n) - R^* \right] = \frac{\text{var}(\mu) \cdot X}{n}$ where $\text{var}(\mu)$ refers to the variance term.

4. Step 4: Generalizing the Expression
   The equation becomes: $P\left\{ R(X_n) - R^* > \varepsilon \right\} \leq \frac{\text{var}(\mu) \cdot X}{n \cdot \varepsilon}$

5. Step 5: Determining the Conditions for $\delta$
   To show the equation holds for any $\delta > 0$, we further proceed as follows: $n \geq \frac{\text{var}(\mu) \cdot X}{\varepsilon \cdot \left( \frac{1}{\delta} \right)} = \frac{\text{var}(\mu) \cdot X}{\varepsilon \cdot \delta}$ Hence, if the condition $n \geq \frac{\text{var}(\mu) \cdot X}{\varepsilon \cdot \delta}$ is satisfied, the probability will hold.

Thus, equation (11) asserts that for any $\varepsilon > 0$ and $\delta > 0$, the inequality holds as long as $n$ satisfies this condition.

Would you like further clarification on any specific steps or additional details? Here are some related questions that might help deepen the understanding of this result:

1. How does the variance affect the upper bound on the probability?
2. Can Markov’s inequality be applied to other forms of random variables?
3. What is the role of the parameter $X$ in this context?
4. How do you interpret the equation in terms of the convergence of $R(X_n)$?
5. Could a different inequality (like Chebyshev's inequality) lead to a tighter bound?

Tip: In practical problems, choosing a larger $n$ can reduce the upper bound on the probability, which leads to more accurate results as the sample size grows.