• Infinitely Large Sequence:

A sequence $\{a_n\}$ is called infinitely large if $\forall K \in \mathbb{R}$ $\exists n_{k} \in \mathbb{N}$ such that $|a_n|>K$ $\forall n\geq n_k$.

Understanding What this means: This means you have a sequence and you are going for larger and larger terms for example you have a sequence $\{a_n\}$ now, let's say you choose $2024 \in \mathbb{R}$ then as the sequence is getting larger and larger after a certain time you must find a term $a_N$ such that $a_N>2024.$
Now, it may be two sided or one sided infinity as we go along the sequence and that is to say the sequence can either go to $\infty$ or $-\infty$ or both simultaneosly.

Why we need this? You may wonder that wtf? We can simply say: $$\lim_{n \to \infty}a_n=\infty$$ But then why this kid of gargons?
The answer is tricky but simple concept of analysis that you must understand.
The reason is how you know what is infinity? This is not a number right! Do you know what number is infinity? Then how can you invoke it in mathematics? You cann't do any mathematical operations with it. Then you can't techmically write some real number $x=\infty$. This is wrong. That's why we need to bring the actual meaning of $\infty$ which is so large that we can't bound it by any real number say $K$. That's why this is said that $\nexists K \in \mathbb{R}$ such that $\infty \leq K$ or $\forall K \in \mathbb{R}$, $K>\infty$.

Velocity: Lets define the velocity of a sequence by $$v=\frac{K}{n_K}$$ Example1: Now, for the following sequence $\{a_n\}=\{1,2,3,4,5, \dots\}$ we have for any $K \in \mathbb{R}$, $n_K= \lfloor K \rfloor+1$. ( Verify ).
Now, $v=\frac{K}{\lfloor K \rfloor+1}$.

Example2: Consider the sequence: $$a_1=1 \\ a_2=1+ \frac{1}{2} \\ a_3=1+ \frac{1}{2}+\frac{1}{3} \\ a_4=1+ \frac{1}{2} +\frac{1}{3} + \frac{1}{4}\\ \vdots \\ a_n=1+ \frac{1}{2} +\frac{1}{3} + \frac{1}{4}+ \dots +\frac{1}{n}\\$$

Velocity Calculation: Observe that, $$a_n=1+ \frac{1}{2} +\frac{1}{3} + \frac{1}{4}+ \dots +\frac{1}{n} =1+ (\frac{1}{2}) +(\frac{1}{3} + \frac{1}{4})+ \dots +\frac{1}{n}\\ >1+ \frac{1}{2}+(\frac{1}{4}+\frac{1}{4})+\dots + \frac{1}{n}\\ =1+\frac{1}{2}+\frac{1}{2}+\dots+\frac{1}{n}$$ So, for $K \in \mathbb{R}$ we need $n_K\geq2^{K}$.
Then $v\leq \frac{K}{2^K}$

Now, compare $v_1=\frac{K}{\lfloor K \rfloor+1}$ and $v_2\leq \frac{K}{2^K}$
Cleraly, $v_1>>v_2$.
Moral: $v_2 << v_1$ but it is steady, so gradually it blows to $\infty$.

• Infinitely Small Sequences:

Definition: A sequence $\{a_n\}$ is called infinitely small if $$\lim_{n \to \infty}a_n=0$$ that is, $|a_n|<\varepsilon$ $\forall n\geq \eta_\varepsilon$.

Question: Try to think of how to interpret it as before? Can you see some similarity bewteen infinitly small and infinitely large sequences?
Try to think by your own then check answer.

• Cauchy Sequences:

ClassNotes

Definition: A sequence $\{a_n\}$ is called cauchy iff $\forall \varepsilon>0$, $\exists K \in \mathbb{N}$ such that $\forall m,n \geq K$
$|a_n-a_m|<\varepsilon$.

Notice: If we fix $m=K$ then $|a_n-a_K|<\varepsilon$ $\forall n\geq K$.
This implies that cauchy sequence is bounded.

Also Notice: Cauchy sequence cann't oscillate.
Proof: Let $\{a_n\}$ be a cauchy sequence. Now, let it be oscillatory(If possible). Now, let $\{a_{n_k}\}$ be a convergent subsequence of $\{a_n\}$. Then we have, $$\lim_{k \to \infty}a_{n_k}=l$$
then by definiton of limit, for $\frac{\varepsilon}{2}$ we have a $N_1 \in \mathbb{N}$ such that
$\forall n_k \geq N_1 \implies$ $|a_{n_k}-l|<\frac{\varepsilon}{2}$....(1)

As $\{a_n\}$ is cauchy, we have a $N_2 \in \mathbb{N}$ such that
$\forall m,n\geq N_2 \implies |a_m-a_n|<\frac{\varepsilon}{2}$.......(2)

Consider $max\{N_1,N_2\}=N$, then equation (1) and (2) holds togethre, and giving us something interesting.
We can think $m=n_k$ now and can say $\forall n_k and n \geq N$ we have, $|a_{n_k}-a_n|<\frac{\varepsilon}{2}$. Now, adding This with equation (1) we get $|a_{n_k}-l|+|a_{n_k}-a_n|<\varepsilon$.
By triangluar inequality we get,
$$|a_n-l| \leq |a_{n_k}-l|+|a_{n_k}-a_n|<\varepsilon$$ And here we get by definiton of limit $$\lim_{n \to \infty}a_n=l$$

So, moral is oscillation simply means convergence and cauchy sequence can't diverge. Together we get, only way a cauchy sequence can't converge is the point of convergence is not inside the set.

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Question: Find an example of cauchy sequence.[Hint: Use a set that don't have GLB or LUB].
Question: Why the hint works?
Question: Prove that a covergent sequence is always cauchy in the real analysis ofcourse.(not in general. forget about the thing, I just wrote it so that when you grow up and read much deeper about analysis and you might came back here and start juding my knowledge.)

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