Real Analysis

Let $(a_n)$ be an increasing sequence. Prove that if $(a_n)$ is unbounded, then it diverges to positive infinity.

Let $(a_n)$ be a decreasing sequence. Prove that if $(a_n)$ is unbounded, then it diverges to negative infinity.

Prove that if an increasing sequence is bounded, then it converges to the supremum of the set of values that it takes on.

Prove that if a decreasing sequence is bounded, then it converges to the infimum of the set of values that it takes on.

Let $a$ be a set. We say that a family of sets is a cover of $a$ if $a$ is a subset of the union of that family of sets.

Find a finite subcover for the given open cover of an interval, if possible.

Prove that a given set is not compact by finding an open cover with no finite subcover.

Prove the classic theorem that says a subset $K$ of $\mathbb{R}$ is compact if and only if it is closed and bounded.

Given a series, determine whether it converges or diverges using the limit comparison test.

Prove that the sum from $N=1$ to $\infty$ of $\frac{1}{N^3 + 4}$ is convergent or divergent using the comparison test.

Prove that the sum from $N=1$ to $\infty$ of $\frac{4^N}{5^N + 8}$ is convergent or divergent using the comparison test.

Demonstrate that a set in $\mathbb{R}$ is disconnected if open sets $U_1$ and $U_2$ can be found such that the intersection of $e$ with both $U_1$ and $U_2$ are non-empty, and their union covers all of $e$, while the intersection of these parts is empty.

Determine if a punctured interval is a disconnected set by identifying appropriate open sets.

Illustrate the disconnection of two sets in bb{R}^2 using open sets to separate them.

Determine if the set $(-2, -1) \cup (-1, 0)$ is connected or disconnected.

Define when a set is considered connected in terms of open sets.

Show that all even numbers are countable by mapping them onto the set of natural numbers.

Show that the set {$\frac{1}{1^2}$, $\frac{1}{2^2}$, $\frac{1}{3^2}$, ...} is countable by creating a bijection to the natural numbers.

Prove that the set of all real numbers between 0 and 1 is uncountable.

Determine whether a given set is countable by demonstrating a one-to-one correspondence with the positive integers.