Real Analysis

Prove the Mean Value Theorem.

Define a recursive sequence $a_1 = \sqrt{2}$ and $a_{n+1} = \sqrt{2 + a_n}$ for $n \geq 1$. Prove that the sequence is monotonic and bounded using the Monotone Sequence Theorem, and find its limit.

Show that the sequence defined by $a_1 = 1$ and $a_{n+1} = 3 - \frac{1}{a_n}$ is increasing and $a_n < 3$ for all $n$. Deduce that the sequence $a_n$ is convergent and find its limit.

Determine if the set of integers is open, closed, or compact.

Determine if the set is open, closed, or compact: the union of the reciprocals of natural numbers with the set containing zero.

Determine if the set of real numbers is open, closed, or compact.

Determine if the set is open, closed, or compact: the union of an open interval (0, 1) and a closed interval [3, 4].

Determine if the set of rational numbers is open, closed, or compact.

Determine if a singleton set containing only the number seven is open, closed, or compact.

Determine if the interval $[1, 3)$ is an open set in the metric space of real numbers defined as $x = \{1 \leq x < 3\} \cup \{x > 4\}$, where the metric is the standard distance function.

Determine the boundary of the set $C = [1, 2]$ within the real number metric space, where the metric is the standard distance function.

Explain the difference between pointwise and uniform convergence of a sequence of functions.

Given a sequence of continuous functions $f_n$, if their pointwise limit $f$ is not continuous, explain why uniform convergence fails.

Demonstrate why a discontinuous function cannot be uniformly approximated by continuous functions.

Consider the sequence of functions $F_n(x) = x^n$ on the interval $[0,1]$. Determine whether this sequence converges pointwise or uniformly, and identify the limit function.

Prove that the sequence $a_n$ of positive numbers defined by the recursive formula $a_1 = 1$ and $a_{n+1} = 3 - \frac{1}{a_n}$ is monotonically increasing, i.e., for all natural numbers $n$, $a_n < a_{n+1}$.

Use induction to prove that the sequence $y_n > -6$ for all $n \in \mathbb{N}$.

Use induction to show that the sequence $y_1, y_2, y_3, \ldots$ is decreasing, i.e., $y_1 \geq y_2 \geq y_3 \geq \cdots$.

Show that $(A_1 \cup A_2 \cup \cdots \cup A_n)^c = A_1^c \cap A_2^c \cap \cdots \cap A_n^c$ for any finite $n$. Use induction to prove it.

Prove that the function $f(x) = \frac{1}{x}$ is not uniformly continuous on the interval (0,1).