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Exploring Real Analysis: Problems and Solutions

Dive into the beauty of real analysis with these problems and solutions. Each problem represents a key concept in the field. Let's explore!

1. Sequences and Series

Problem:

Let \( (a_n) \) be a sequence defined by \( a_1 = 1 \) and \( a_{n+1} = \frac{a_n + 2}{2} \). Prove that \( (a_n) \) converges and find its limit.

Solution:

The sequence \( (a_n) \) is bounded and monotonic. Calculating:

        \( a_1 = 1, a_2 = 1.5, a_3 = 1.75, \dots \)
    

Assuming convergence to \( L \), we solve: \[ L = \frac{L + 2}{2} \implies L = 2 \]

Thus, \( (a_n) \) converges to 2.

2. Limits and Continuity

Problem:

Prove that \( f(x) = \frac{1}{x} \) is uniformly continuous on \( [1, \infty) \), but not on \( (0, \infty) \).

Solution:

On \( [1, \infty) \), for \( \epsilon > 0 \), choose \( \delta = \epsilon \). Then: \[ |x - y| < \delta \implies \left| \frac{1}{x} - \frac{1}{y} \right| \leq \epsilon \]

On \( (0, \infty) \), as \( x \to 0^+ \), \( \frac{1}{x} \to \infty \), so it fails uniform continuity.

3. Differentiation

Problem:

Let \( f(x) = x^2 \sin\left(\frac{1}{x}\right) \) for \( x \neq 0 \) and \( f(0) = 0 \). Prove whether \( f(x) \) is differentiable at \( x = 0 \).

Solution:

For \( x \neq 0 \), the derivative is: \[ f'(x) = 2x\sin\left(\frac{1}{x}\right) - \cos\left(\frac{1}{x}\right) \]

At \( x = 0 \), compute: \[ \lim_{x \to 0} \frac{f(x) - f(0)}{x} = 0 \]

Hence, \( f \) is differentiable at \( x = 0 \), with \( f'(0) = 0 \).

4. Integration

Problem:

Evaluate whether the improper integral \( \int_1^\infty \frac{\sin x}{x} \, dx \) converges.

Solution:

The integral converges conditionally. Using Dirichlet's test: \[ \text{Oscillatory } \sin x \text{ combined with the decreasing } \frac{1}{x}. \]

5. Metric Spaces

Problem:

Prove that any closed and bounded subset of \( \mathbb{R} \) is compact.

Solution:

By the Heine-Borel theorem, closed and bounded subsets of \( \mathbb{R} \) are compact. Proof hinges on every sequence in the set having a convergent subsequence.