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.