[b](a)[/b] Let $n$ be a positive integer, prove that
\[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\]

[b](b)[/b] Find a positive integer $n$ for which
\[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} +  \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]
### Part (a)
To prove:
\[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt{n}} \]

1. **Rationalize the left-hand side:**
   \[
   \sqrt{n+1} - \sqrt{n} = \frac{(\sqrt{n+1} - \sqrt{n})(\sqrt{n+1} + \sqrt{n})}{\sqrt{n+1} + \sqrt{n}} = \frac{(n+1) - n}{\sqrt{n+1} + \sqrt{n}} = \frac{1}{\sqrt{n+1} + \sqrt{n}}
   \]

2. **Compare the rationalized form with the right-hand side:**
   \[
   \frac{1}{\sqrt{n+1} + \sqrt{n}} < \frac{1}{2 \sqrt{n}}
   \]

3. **Simplify the inequality:**
   \[
   \sqrt{n+1} + \sqrt{n} > 2 \sqrt{n}
   \]

4. **Square both sides to eliminate the square roots:**
   \[
   (\sqrt{n+1} + \sqrt{n})^2 > (2 \sqrt{n})^2
   \]
   \[
   n + 1 + 2\sqrt{n(n+1)} + n > 4n
   \]
   \[
   2n + 1 + 2\sqrt{n(n+1)} > 4n
   \]
   \[
   1 + 2\sqrt{n(n+1)} > 2n
   \]
   \[
   1 > 2n - 2\sqrt{n(n+1)}
   \]
   \[
   1 > 2n - 2\sqrt{n^2 + n}
   \]
   \[
   1 > 2n - 2n\sqrt{1 + \frac{1}{n}}
   \]

5. **Simplify further:**
   \[
   1 > 2n(1 - \sqrt{1 + \frac{1}{n}})
   \]
   Since \(\sqrt{1 + \frac{1}{n}} < 1 + \frac{1}{2n}\) for large \(n\), we have:
   \[
   1 > 2n(1 - (1 + \frac{1}{2n}))
   \]
   \[
   1 > 2n(1 - 1 - \frac{1}{2n})
   \]
   \[
   1 > 2n(-\frac{1}{2n})
   \]
   \[
   1 > -1
   \]
   This is always true, thus proving the inequality.

### Part (b)
To find \( n \) such that:
\[ \left\lfloor 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} + \cdots + \frac{1}{\sqrt{n}} \right\rfloor = 12 \]

1. **Define the sum \( A \):**
   \[
   A = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} + \cdots + \frac{1}{\sqrt{n}}
   \]

2. **Use the inequality from part (a):**
   \[
   \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt{n}}
   \]

3. **Approximate the sum \( A \):**
   \[
   A \approx 2(\sqrt{n+1} - 1)
   \]

4. **Set up the inequality:**
   \[
   12 \leq 2(\sqrt{n+1} - 1) < 13
   \]
   \[
   6 \leq \sqrt{n+1} - 1 < 6.5
   \]
   \[
   7 \leq \sqrt{n+1} < 7.5
   \]
   \[
   49 \leq n+1 < 56.25
   \]
   \[
   48 \leq n < 55.25
   \]

5. **Determine the integer values of \( n \):**
   \[
   n \in \{48, 49, 50, 51, 52, 53, 54, 55\}
   \]

The final answer is \( \boxed{ n \in \{48, 49, 50, 51, 52, 53, 54, 55\} } \)