3:37 AM
Hi. Could you tell us why we are considering m<n in this math.stackexchange.com/questions/1340333/…
Why is this enough?
we could, for example, choose a vector U that includes more than m terms

4 hours later…
8:06 AM
@Maxxx Am I right in assuming that you wanted to link to this answer (rather than to a comment): Hamel Basis in Infinite dimensional Banach Space without Baire Category Theorem
The goal of that part of the proof is to show that $\lim u_n\ne u$. (For the given sequence $u_n$.)
Clearly, for this it suffices to know about behavior of $u_n$ for large enough $n$.
The number $m$ is uniquely determined by the vector $u$.
Since $\{e_n; n\in\mathbb N\}$ is a Hamel basis, every vector from $X$ has exactly one expression as a finite linear combination of elements from the basis.

8:31 AM
We can continue this discussion here - after all it is related to my answer. But maybe it might be interesting for you to know that there is a chatroom for functional analysis and related areas. (Since you are asking about this specific problem, you might be interested in functional analysis.)
I will also add that the proof of the same fact based on Baire category theorem can be found here: Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

5 hours later…
1:15 PM
Kislay Tripathi, you have relatively quickly deleted your question on meta: How to search a question on mathstackexchange.
Almost immediately after I've added the comment: This question seems rather similar to: How to ask the question which aren't ask before and How to avoid duplications?.
Probably if you look at those questions, you can find some reasonable advice about searching.
I will add a few words.
I will stress that: Duplicates can be useful.
You have mentioned that you worry about a bit:
> It puts me on the risk of being restricted from asking any question.
IIRC duplicates contribute much less to question ban than downvotes and deleted questions.
Since you specifically mentioned searching for formulas, you might try Approach0.
There is also a chatroom dedicated to searching - you can also ask there when looking for some question. (Although I will admitted that not many users visit that room.)
I will also add that your recent question Summation involving integral would be closed even if it wasn't duplicate. You can see in the revision history that close reason was off-topic.
To be more specific, it was closed as off-topic/missing context. (Although such details are not displayed in revision history.)
I think I even left under that question a comment with link to Approach0 search. (However, comments were cleaned-up after merging.)
If you feel like you need more advice about searching, you can ask here - and if I am around and have a bit of time I'll try to respond. It is probably even better to ask in this chat room: In the search of a question. It is chatroom dedicated to searching.

2 hours later…
3:24 PM
Hi @Maxxx. Was the above explanation satisfactory or is there still some problem with this argument?

4:20 PM
Hi @Martin Sleziak. Everything is very good! Thank you. I forgot that only finite linear combinations are needed

Ok, I am glad that is cleared up.