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To solve this problem, we need to analyze the function \( f(x) = e^x - mx + n - 1 \) and ensure that \( f(x) \geq 0 \) for all \( x \in \mathbb{R} \).

Given \( f(x) \geq 0 \), the function should not have any roots that would make it negative. We will analyze the conditions under which this happens.

1. **Function Analysis**:
   \( f(x) = e^x - mx + n - 1 \)

   For \( f(x) \geq 0 \) for all \( x \), the function must not dip below the x-axis.

2. **Critical Points and Derivatives**:
   Let's find the first derivative \( f'(x) \):
   \[
   f'(x) = \frac{d}{dx}(e^x - mx + n - 1) = e^x - m
   \]
   The critical points occur when \( f'(x) = 0 \):
   \[
   e^x - m = 0 \Rightarrow e^x = m \Rightarrow x = \ln m
   \]

3. **Second Derivative Test**:
   \[
   f''(x) = \frac{d}{dx}(e^x - m) = e^x
   \]
   Since \( e^x \) is always positive, \( f(x) \) has a local minimum at \( x = \ln m \).

4. **Value at Critical Point**:
   Evaluate \( f(x) \) at \( x = \ln m \):
   \[
   f(\ln m) = e^{\ln m} - m(\ln m) + n - 1 = m - m \ln m + n - 1
   \]
   For \( f(x) \geq 0 \) for all \( x \):
   \[
   m - m \ln m + n - 1 \geq 0 \Rightarrow m - m \ln m + n \geq 1 \Rightarrow n \geq m \ln m - m + 1
   \]

5. **Finding \( \frac{n - m}{m} \)**:
   To find the minimum value of \( \frac{n - m}{m} \):
   \[
   \frac{n - m}{m} \geq \frac{m \ln m - m + 1 - m}{m} = \frac{m \ln m - 2m + 1}{m} = \ln m - 2 + \frac{1}{m}
   \]
   The minimum value of \( \frac{n - m}{m} \) occurs when this expression is minimized.

6. **Minimizing the Expression**:
   To minimize \( \ln m - 2 + \frac{1}{m} \), we need to analyze this expression. By setting the derivative to zero and solving, it can be shown that the minimum value occurs at \( m = 1 \).

7. **Substitute \( m = 1 \)**:
   \[
   \ln 1 - 2 + \frac{1}{1} = 0 - 2 + 1 = -1
   \]

Thus, the minimum value of \( \frac{n - m}{m} \) is \(-1\).

Would you like more details or have any questions?

Here are five more questions you can ask:
1. Can you explain how to find the critical points of a function?
2. What is the second derivative test and how is it used?
3. How do we determine the minimum value of a function?
4. Can you show the steps to solve a similar inequality problem?
5. What are the applications of these kinds of optimization problems?

**Tip:** Always verify the conditions of the problem and ensure all constraints are satisfied before finalizing your answer.

Learn how to analyze and solve the function f(x) = e^x - mx + n - 1 to ensure it remains non-negative for all x. This problem involves examining exponential functions, derivatives, and inequalities. Suitable for college-level mathematics.

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