Lecture 4: Foundations of Limits

4.1 Basic Concepts of Limits

• Mainly two types
• 1. Sequence limits;
• 2. Function limit;

4.1.1 Limits of Sequences

#####Definition: Sequencelimit

Description: $\lim_{n\to\infty}x_n=a$ iff, for every $\varepsilon>0$, there is an $N$ such that $n>N$ implies $|x_n-a|<\varepsilon$.

Explanation

• $\varepsilon>0$: The proximity of the function;
• Geometric meaning: $x_n$ within a small area before and after $\varepsilon$;
• $N$: whenever $n>N$, every term satisfies $|x_n-a|<\varepsilon$; only finitely many terms precede $N$.
• Finite terms are always bounded;
• The limit of sequence $\left\{x_n\right\}$ is independent of the previous finite term;

Concept: Conclusion 1

• $$\lim_{n\to\infty}x_n=a\Leftrightarrow\lim_{k\to\infty}x_{2k-1}\overset{\color{red}{}}{\operatorname*{=}}\lim_{k\to\infty}x_{2k}=a.$$

-If a sequence has a limit, then all its subsequences have a limit;

Concept: Conclusion 2

• 1. $$\text{ If }\lim_{n\to\infty}x_n=a,\text{ then }\lim_{n\to\infty}\lvert x_n\rvert=\lvert a\rvert,\text{ But the opposite is not true }$$
• 2. $$\lim_{n\to\infty}x_n=0\text{ The necessary and sufficient condition for this is: }\lim_{n\to\infty}|x_n|=\mathbf{0}$$

Concept: Conclusion 3

• If the subsequence of a sequence satisfies the following two conditions, the value of the complete sequence can be derived:
• 1. All values in the subsequence are equal;
• 2. Several subsequences have taken all possible scenarios from the original sequence;