I am studying analysis 1 - so this is still some extremely beginner stuff. So please keep the terms in the most simple way possible, thank you.
If we were to have a sequence, lets say: 3, 1, 3, 4, 3 , 7, 3 , 10. Where every other element grows up to infinity. Would this sequence be considered as going up to infinity. My professor says no. If so, why?
Also, follow up question. Because it's too similar to this one:
If we were to have a sequence, let's say 1,2,4,3,7,4,10,5. Where odd numbered index element has +3 added from the previous odd numbered one, and evey even has +1. Would this sequence be considered as going to infinity? Professor said that yes.
Finally, do either of these sequences have a limit?
Maths is all about precision. When you say a sequence goes to infinity or to some finite number, what exactly do you mean? How would you define "going to" in mathematical terms? These are the things where you should start analysis. Instead of looking for definition, you can may be try to think of a logical definition on your own and then compare it to the standard definition.
Here is the standard definition:
Given a sequence $\{x_n\}$, we say it converges to some $a$ if for any $\epsilon>0$, $\exists$ $N\in\mathbb N$ such that $|x_n - a|<\epsilon$ for all $n>N$.
As with any definition in mathematics, instead of getting lost into the notations, you should try to understand what it means. That $|x_n - a|<\epsilon$ means that the difference between $x_n$ and $a$ is smaller than $\epsilon$ but now notice "any" $\epsilon>0$ in the definition. It essentially means that no matter how small the $\epsilon$ is (say as small as $10^{-100}$) you can always find a natural number $N$ such that beyond $N$, the difference between $x_n$'s and $a$ becomes smaller than $\epsilon$.
At this point, it is good to look at an example. Take the sequence $0.1$, $0.01$, $0.001 \ldots$ which you can also express as $\{x_n = 10^{-n}\}$ as $n$ goes from $1$ to $\infty$. We say this sequence goes to $0$ because if we take any $\epsilon>0$, for example $\epsilon = 10^{-10}$ and you can see that as you go beyond $n=10$, the difference between $x_n$ and $0$ is smaller than $\epsilon$. Take $\epsilon = 10^{-100}$, the difference gets smaller than $\epsilon$ beyond $n = 100$. Take $\epsilon = 10^{-1000}$, the difference gets smaller beyond $n = 1000$. It simply doesn't matter how small $\epsilon$ becomes, we can eventually find a number beyond which the difference gets smaller than $\epsilon$. This is the precise formulation of "going to" or what we call "convergence".
Now take another sequence $1$, $0.1$, $1$, $0.01$, $1$, $0.001\ldots$. Can you say this converges to $0$? It doesn't for take $\epsilon = 0.5$. Can you find a number beyond which the difference gets smaller than $0.5$? No, because no matter how far ahead you go into the sequence, there is always a term equal to $1$ and $1 - 0 \not< 0.5$, is it?
Now coming back to your problem, you are considering going to infinity which needs a slight modification into the definition I gave. But I suggest you look for that definition yourself and then compare it to this and think why it is the way it is and what's the reason behind those small modifications? Finally, test your examples with that definition and you will get all your answers.