# Lecture - Analysis VII

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/7/ · Updated: 2022-11-05 · Tags: uni, lecture

- [Course - Analysis MT22](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/)

### Flashcards
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 $b_n \to M$?::
$$
\frac{a_n}{b_n} \to \frac{L}{M} \text{ for } M\ne 0
$$

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