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The problem provided asks to prove the inequality for all natural numbers \( n \):

\[
1 + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} \leq 2 - \frac{1}{n}.
\]

### Steps to prove the inequality:

1. **Express the sum**:
   The sum on the left-hand side is a partial sum of the series:
   \[
   S_n = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2}.
   \]
   This is the sum of the first \( n \) terms of the series:
   \[
   \sum_{k=1}^{n} \frac{1}{k^2}.
   \]

2. **Upper Bound of the Series**:
   It is known that the sum of the infinite series:
   \[
   \sum_{k=1}^{\infty} \frac{1}{k^2} = \zeta(2) = \frac{\pi^2}{6}.
   \]
   Numerically, \( \zeta(2) \approx 1.64493 \), which is less than 2.

   So, for all \( n \), we can already see that the infinite sum is bounded by a constant less than 2. However, we are interested in bounding the partial sum \( S_n \).

3. **Analysis of the Bound**:
   The given inequality suggests that for each natural number \( n \), the partial sum \( S_n \) is bounded by:
   \[
   S_n \leq 2 - \frac{1}{n}.
   \]
   Since the series converges to \( \frac{\pi^2}{6} \) as \( n \to \infty \), and \( 2 - \frac{1}{n} \to 2 \), this is reasonable because the partial sum should approach a limit less than 2, but it must also be corrected for the decreasing size of the terms.

4. **Induction Hypothesis**:
   To formally prove the inequality, we can use mathematical induction or comparison of the terms of the series with decreasing fractions, showing that the difference between the sum and 2 becomes smaller as \( n \) increases.

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Would you like a more detailed walkthrough of each step, including an induction proof or comparison analysis?

Here are 5 questions that might deepen your understanding:
1. What is the significance of the convergence of the infinite sum \( \zeta(2) \) in this inequality?
2. How does the term \( \frac{1}{n} \) act as a correction factor in the inequality?
3. Could a more precise upper bound for the partial sum \( S_n \) be computed?
4. How does the difference between \( S_n \) and \( \frac{\pi^2}{6} \) behave for large \( n \)?
5. What other techniques could be used to prove inequalities involving sums of series?

**Tip**: For series and inequalities, combining known results from convergence can simplify proofs significantly.

Prove the inequality: For all natural numbers n, 1 + 1/2² + 1/3² + … + 1/n² ≤ 2 - 1/n.

Explore how to prove the inequality involving the partial sum of the series 1 + 1/2² + 1/3² + ... + 1/n², showing that it is bounded by 2 - 1/n. This proof utilizes known results from the convergence of series and the application of induction or comparison.

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