How freshman Cauchy's definition has reduced

Under the cut, I will tell you a small and not a shocking story, most of you will probably say that I thought up at least some importance of this event and that everything described below is obvious, but for me it was a small victory. If still interesting, welcome.

Immediately I would like to discuss a few points: I am a freshman, so I don’t claim anything for any discovery in mathematical analysis and wrote an article to listen to the opinion of local experts.

It all started with the first in my life colloquium on the discipline Mathematical analysis, one of the tasks of which contained the definition of a non-fundamental Cauchy sequence. Under the cut traffic.
I, without hesitation, wrote the following:

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Looked, figured and left. A week later I got my job with an unread number with a brief explanation.

It upset me, and I decided to understand whether my version was wrong, I decided to approach the teacher. After lengthy discussions and formal explanations, I asked me to give me a counter-example, for which I received consent, but the teacher said that he needed time and that even if we cannot come up with such an example does not mean that it does not exist.

After a couple of years, the teacher called me to prove to me a counter-example that my definition is only a special case. At that time, I myself was already inclined to this, but decided to listen. After he painted a huge and complex sequence, which your humble servant, alas, forgot, he began to explain to me and here I understand that this example more than fully satisfies my definition of a non-fundamental sequence. Already at this stage, my assessment was corrected to 5, with the objection that the teacher was nevertheless convinced that formally I was wrong. However, no one thought about evaluation, I spent the whole week thinking about a counter-example for my definition.

A week later, I did not come to anything, I approached the teacher and told that I had failed in the search for an anti-example, I heard that according to the teacher, both statements are equivalent.
Here is the proof we made:
We will prove the equivalence of assertions of fundamentality. Take the negation of my definition of a non-fundamental sequence.

And consider the two following to prove equivalence.


Thus, k-zero can be removed from the definition of Cauchy. I would like to receive feedback, especially on the subject of correctness of the proof.