Lecture - Analysis MT22, VII

How would you write that $a _ n \to \infty$ as $n \to \infty$?

\[\forall M \in \mathbb{R} \space \exists N \text{ s.t. } \forall n > N, a_n > M\]

How would you write that $a _ n \to -\infty$ as $n \to \infty$?

\[\forall M \in \mathbb{R} \space \exists N \text{ s.t. } \forall n > N, a_n < M\]

What is the definition of a subsequence of $(a _ n)$?

Let $(n _ r)$ be a strictly increasing sequence of natural numbers. Then $(a _ {n _ r})$ is a subsequence of $(a _ n)$.

If $(a _ n) \to L$ as $n \to \infty$, what can you say about the convergence of subsequences?

\[(a_{n_r}) \to L\]

What can you say about the convergence of $(a _ n)$ if two of its subsequences converge to different limits?

$(a _ n)$ is divergent.

How many theorems are there overall about the algebra of limits?

8

What is the “algebra of limits” theorem about constant sequences?

If $a _ n = c$ for all $n \ge 1$ then $a _ n \to c$ as $n \to \infty$.

What is the “algebra of limits” theorem about about sums for limits $a _ n \to L$ and $b _ n \to M$?

\[a_n + b_n \to L + M\]

What is the “algebra of limits” theorem about about scalar multiples for limits $a _ n \to L$ ?

\[ca_n \to cL\]

What is the “algebra of limits” theorem about about differences for limits $a _ n \to L$ and $b _ n \to M$?

\[a_n - b_n \to L - M\]

What is the “algebra of limits” theorem about about the modulus for limits $a _ n \to L$?

\[|a_n| \to |L|\]

What is the “algebra of limits” theorem about about products for limits $a _ n \to L$ and $b _ n \to M$?

\[a_n b_n \to LM\]

What is the “algebra of limits” theorem about about reciprocals for limits $b _ n \to M$?

\[\frac{1}{b_n} \to \frac{1}{M} \text{ for } M \ne 0\]

What is the “algebra of limits” theorem about about quotients for limits $a _ n \to L$ and $b _ n \to M$?

\[\frac{a_n}{b_n} \to \frac{L}{M} \text{ for } M\ne 0\]