Real Analysis

Calculate the limit as $n \to \infty$ of $\frac{1^a + 1 + 2^a + 1 + \ldots + n^a + 1}{n (1^a + 2^a + \ldots + n^a)}$. Determine the allowable values of $a$.

Evaluate the double integral $\int_{0}^{2} \int_{\sqrt[3]{x}}^{2} \frac{1}{y^4 + 1} \, dy \, dx$ by changing the order of integration.

Use the Intermediate Value Theorem to show that there is a root of the equation $f(x) = x^2 - x - 12$ in the interval [3, 5].

Use the Intermediate Value Theorem to find the value of $c$ in the interval $[1, 4]$ such that $f(c) = 19$, given $f(x) = 2x^2 + 3x + 5$.

Prove the Intermediate Value Theorem.

Prove that the set of rational numbers $\mathbb{Q}$ is countable by using the classical method of defining the height of a rational number and showing it as a countable union of finite sets.

Using Campbell's method with base 11, prove that the set of positive rational numbers $\mathbb{Q}^+$ is countable.

Establish a bijective correspondence between the integers and the rational numbers.

Suppose S is a non-empty subset of the real numbers that is bounded above by M. Then S has a least upper bound, meaning the supremum exists.

Provide an example of a set in the rational numbers that does not have a least upper bound.

Show that the set of rational numbers $Q$ does not have the least upper bound property by considering the set $A = \{ p \in \mathbb{Q} : p^2 < 2 \}$ and demonstrating that the supremum of this set is $\sqrt{2}$, which is not a rational number.

Find the limit points and derive the set of a set $A$ in the topology $\tau$ on a set $X$, where $X = \{ A, B, C, D, E \}$.

Prove that a subset $A$ of a topological space $X$ is closed if and only if it contains all of its limit points.

Let the sequence $s_n$ be this sequence that repeats in a pattern of $0, 1, 2, 1, 0, 1, 2, 1$ and so on. What is the limit superior of this sequence?

Consider the sequence $t_n$ which approaches 3 with its odd terms and approaches 1 from above with its even terms. What is the limit superior of this sequence?

For the sequence $a_n$ whose nth term is $\frac{1 + (-1)^n n}{n}$, what are the limit superior and limit inferior of the sequence?

Let $b_n$ be the sequence $n \times \sin\left( \frac{\pi n}{2} \right)$. What are the limit superior and limit inferior of this sequence?

For a sequence $a_n$, determine the limit superior and limit inferior.

Prove Rolle's Theorem: Suppose $f$ is continuous on a closed interval $[a, b]$ and differentiable on the corresponding open interval $(a, b)$. Show that if $f(a) = f(b)$, then there exists a $c$ in $(a, b)$ such that $f'(c) = 0$.

Prove the Mean Value Theorem: If $f$ is continuous on a closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then there exists a $c$ between $a$ and $b$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.