Real Analysis: Sequences and Convergence

Find all of the accumulation points of the sequence $a_n$ where $a_n = (1 + (-1)^n, 2^n + (-2)^n)$.

Prove that the sequence definition and the neighborhood definition of limit points are equivalent.

Prove that any bounded sequence has a subsequence that converges.

Given a sequence, determine if it is both bounded and monotonic. If it is, prove that it converges.

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.

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 $a$.

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.

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., $y_1 \geq y_2 \geq y_3 \geq \cdots$.

Consider the sequence $a_n = \frac{n}{2}$. Prove that this sequence diverges to positive infinity.

Prove that the sequence $a_n = -\frac{n}{2}$ diverges to negative infinity.

Prove that the sequence $a_n = (-1)^n$ diverges.

Prove that the sequence $\frac{1}{n}$ approaches 0 as $n$ approaches infinity.

Prove the order limit theorem for convergent sequences, including the three statements:

1) If every term of a convergent sequence is at least 0, then the limit of that sequence is also at least 0.

2) If every term of one convergent sequence is less than or equal to some other convergent sequence, then their limits have that same relation.

3) If a convergent sequence is bounded below by some real number, then the limit of that convergent sequence is at least as big as that lower bound. Similarly, if a convergent sequence is bounded above by some real number, then its limit is less than or equal to that real number.

Show that the sequence $a_n = 2 + \frac{1}{n+1}$ converges to a limit $a = 2$.