Real Analysis

Prove that if a sequence $(a_n)$ diverges to infinity, then all of its subsequences also diverge to infinity.

Prove that the sequence of functions $f_n(x) = \frac{x}{1 + nx^2}$ converges uniformly on the set of real numbers to 0.

Consider a sequence of functions $f_n$ that converges pointwise to a function $f$. Demonstrate why this sequence does not exhibit uniform convergence.

Consider the following sentence: 'This statement is false.'

Prove that if $a$ and $b$ are real numbers and $a$ is positive, then there exists a natural number $n$ such that $n \times a > b$.

Prove that for any real number $x$, there exists a natural number greater than $x$, meaning the natural numbers are unbounded above.

Consider the recursively defined sequence $a_{n+1} = \frac{1}{3} a_n + 1$ with $a_1 = 1$. Use the Monotone Convergence Theorem to show that this sequence converges.

Given the recursive sequence defined by $a_1 = 1$ and $a_n = 1 + \frac{1}{1 + a_{n-1}}$ for $n \geq 2$, determine the limit of the sequence as $n \to \infty$.

For the Riemann integral, we always start with a partition of the x-axis. Then, we approximate the area under the function's graph by summing up the areas of rectangles formed by choosing points on the x-axis and using those points to determine the width and height of each rectangle. The main question is whether this method correctly represents the area, and we need to ensure that this method defines the integral accurately, even with different choices of partitions and values.

Let $f$ be the function defined on the closed interval from 0 to 1 by $f(x) = 0$ if $x$ is not a rational number, and $f(x) = 1$ if $x$ is a rational number. Determine if the function is Riemann integrable.

Exercise 1: Suppose you've got a function on the interval from 0 to 1 which is 0 except at finitely many points. Show that $f$ is integrable and the value of the integral is 0.

Exercise 2: Suppose $f$ is the function on the closed interval from 0 to 1 which is defined by $f(x) = 0$ if $x$ is not equal to $\frac{1}{n}$ for any natural number $n$ and $f(x) = 1$ if $x$ is equal to $\frac{1}{n}$ for some natural number $n$. Is this function Riemann integrable?

Find the value of $c$ for the function $f(x) = x^2 - 2x + 1$ on the interval $[-1, 3]$ where $f'(c) = 0$ using Rolle's Theorem.

Evaluate if Rolle's Theorem applies for the function $f(x) = \frac{x^2 - 4}{x}$ on the interval $[-2, 2]$.

Verify Rolle's Theorem and find the value or values of c that satisfy it for the following function on the given interval.

Prove that the sequence $\frac{3n+1}{n+2}$ converges to the limit 3 using the epsilon definition of the limit of a sequence.

Make the quantity $\frac{3n + 2}{2n + 5} - \frac{3}{2}$ less than epsilon for $n > N$ in an epsilon-N proof.

Write all subsets of the set $A$, given $A = \{1, 2, 3, 4\}$.

Write all subsets of the set $B$, given $B = \{ 1, 2, \emptyset \}$.

Write all subsets of the set $C$, given $C = \{ \{\emptyset\} \}$.