Real Analysis

Write all subsets of the empty set D.

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

Write all subsets of the set $F$, given $F = \{ \mathbb{R}, \mathbb{Q}, \mathbb{N} \}$.

Write all subsets of the set $G$, given $G = \{ \mathbb{R}, \{ \mathbb{Q}, \mathbb{N} \} \}$.

Write all subsets of the set $H$, given $H = \{ \{0, 1\}, \{0, 1, \{2\}\}, \{0\} \}$.

Prove that the real numbers and the empty set are open sets.

Explain why a closed interval, such as $[0, 1]$, is not an open set.

Prove that the supremum of the set {$\displaystyle \frac{n}{n+1}$ | $n \in \mathbb{N}$} is 1.

Find the radius and interval of convergence for the Taylor series $\sum_{n=0}^{\infty} x^n$.

Find the Taylor series centered at a given value $a$ and find the associated radius of convergence for $f(x) = e^{2x}$ at $a = 6$.

Prove that the function $x^2$ is uniformly continuous on the closed interval $[-5, 5]$.

Prove that the function $\sin\left(\frac{1}{x}\right)$ is continuous but not uniformly continuous on the open interval $(0, 1)$.

Illustrate the difference between continuous functions and uniformly continuous functions using examples.

Prove that if there exists a constant $M$ and a constant $\alpha$ where $0 \leq \alpha < 1$, such that $|c_k x^k| \leq M \alpha^k$ for all $x$ in a subset $A$ of the domain of convergence, then the power series $\sum_{k=0}^{\infty} c_k x^k$ converges uniformly on $A$.

Prove that the series $\sum_{n=1}^{\infty} \frac{1}{1 + n^2} x^4$ is uniformly convergent on the interval $[1, \infty)$.

Given a function $f(x) = x$ on the interval $[1, 5]$, calculate the lower sum $L(f, P)$ and the upper sum $U(f, P)$ where $P = \{1, \frac{3}{2}, 2, 4, 5\}$.

Approximate the area under the curve of the function $f(x) = 1 + x^2$ on the interval $[-1, 1]$ using 4 subintervals. Determine the lower sum by choosing $x_i^*$ such that $f(x_i^*)$ is minimized within each subinterval, and the upper sum by choosing $x_i^*$ such that $f(x_i^*)$ is maximized within each subinterval.

Determine if the series $
}

converges uniformly.

Check if the sequence of partial sums (SOPS) of a given series of functions is uniformly convergent in the interval [0, 1].

Explain Weierstrass's M-test and its application to determine if a series converges uniformly in a closed interval [a, b].