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The given mathematical proof involves inequalities and simplifications using properties of sequences and roots. Here's a step-by-step explanation of each part of the derivation:

### Step-by-Step Explanation:

1. **Initial Expression**:
   \[
   a^n - 1 = (a - 1) (a^{n-1} + a^{n-2} + \ldots + 1)
   \]
   This identity expresses \( a^n - 1 \) as the product of \( (a - 1) \) and the sum of a geometric series.

2. **Expression for the \( n \)-th Root**:
   \[
   \sqrt[n]{n - 1} = \frac{n - 1}{\frac{n-1}{n} + \frac{n-2}{n} + \ldots + \frac{0}{n}}
   \]
   The \( n \)-th root of \( n - 1 \) is expressed in a form that involves fractions of integers.

3. **Simplifying the Denominator**:
   - The denominator consists of a series of fractions that approximate an arithmetic series.
   - The key inequality:
     \[
     \frac{n - 1}{n} + \frac{n - 2}{n} + \ldots + \frac{0}{n} \leq n \cdot \sqrt{n}
     \]
     This simplifies the previous expression.

4. **Rewriting and Inequality**:
   - This leads to:
     \[
     \frac{n - 1}{n \cdot \sqrt{n}}
     \]
   - The simplification relies on estimates involving the terms approaching \(\frac{1}{\sqrt{n}}\).

5. **Further Simplification**:
   - The approximation continues:
     \[
     \frac{n - 1}{n \cdot n^{\frac{1}{2n}}} < \frac{n^{\frac{1}{2n}}}{\sqrt{n}} \leq \frac{2}{\sqrt{n}}
     \]
   - Each step uses inequalities and approximations to tighten bounds and express the terms in simpler forms.

### Questions for Further Exploration:
1. How does the inequality involving \(\sqrt{n}\) impact the overall simplification?
2. What assumptions are made during the simplifications, and are they always valid?
3. Can this proof be extended to other values of \( a \) or generalized for different series?
4. What role does the geometric series identity play in simplifying this problem?
5. How can we visualize the growth of the terms involved to better understand the bounds?

### Tip:
When dealing with inequalities, always ensure the bounds are justified, especially when approximating sums or products of series.

Explain each step of the given mathematical proof.

This page explains a step-by-step mathematical proof involving inequalities and the simplification of series and roots. It covers concepts such as the geometric series theorem, inequalities, and n-th root approximations, suitable for advanced high school or undergraduate students.

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