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The question asks to prove the inequality:

\[
\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{n}} > \sqrt{n}
\]
for all \( n \in \mathbb{N}, n > 1 \).

### Proof (by induction):

We will use mathematical induction to prove this inequality.

**Step 1: Base case for \( n = 2 \)**

For \( n = 2 \), we calculate both sides of the inequality.

The left-hand side (LHS) is:

\[
\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} = 1 + \frac{1}{\sqrt{2}} \approx 1 + 0.707 \approx 1.707
\]

The right-hand side (RHS) is:

\[
\sqrt{2} \approx 1.414
\]

Clearly, \( 1.707 > 1.414 \), so the inequality holds for \( n = 2 \).

**Step 2: Inductive hypothesis**

Assume the inequality holds for some \( n = k \), i.e.,

\[
\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{k}} > \sqrt{k}.
\]

**Step 3: Inductive step**

We need to prove that the inequality holds for \( n = k+1 \), i.e.,

\[
\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{k}} + \frac{1}{\sqrt{k+1}} > \sqrt{k+1}.
\]

From the inductive hypothesis, we know:

\[
\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{k}} > \sqrt{k}.
\]

Adding \( \frac{1}{\sqrt{k+1}} \) to both sides gives:

\[
\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \dots + \frac{1}{\sqrt{k}} + \frac{1}{\sqrt{k+1}} > \sqrt{k} + \frac{1}{\sqrt{k+1}}.
\]

Now, we need to show that:

\[
\sqrt{k} + \frac{1}{\sqrt{k+1}} > \sqrt{k+1}.
\]

We can analyze this by examining the approximation:

For large \( k \), \( \sqrt{k+1} \) is approximately \( \sqrt{k} + \frac{1}{2\sqrt{k}} \), meaning:

\[
\sqrt{k+1} \approx \sqrt{k} + \frac{1}{2\sqrt{k}}.
\]

Since \( \frac{1}{\sqrt{k+1}} \) is greater than \( \frac{1}{2\sqrt{k}} \), we have:

\[
\sqrt{k} + \frac{1}{\sqrt{k+1}} > \sqrt{k+1}.
\]

Thus, the inequality holds for \( n = k+1 \).

By the principle of mathematical induction, the inequality is true for all \( n \geq 2 \).

---

Would you like further details or clarifications?

Here are some related questions:
1. How does the approximation \( \sqrt{k+1} \approx \sqrt{k} + \frac{1}{2\sqrt{k}} \) work?
2. What other methods could be used to prove this inequality?
3. Can we explore how the sum behaves for very large \( n \)?
4. How do we rigorously prove that \( \frac{1}{\sqrt{k+1}} > \frac{1}{2\sqrt{k}} \)?
5. Can this inequality be applied in any practical scenario, such as estimating large sums?

**Tip:** Induction proofs often require establishing a clear and correct inductive step. Always verify your assumptions hold throughout the process.

Prove that 1/√1 + 1/√2 + ... + 1/√n > √n for all n ∈ ℕ, n > 1.

Explore the detailed proof using mathematical induction for the inequality 1/√1 + 1/√2 + ... + 1/√n > √n. This proof covers key concepts in series and inequalities, suitable for advanced high school or early college students.

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