How can sequences satisfy vector space properties? If sequence is not a vector. How can sequences satisfy vector space properties? If sequence is not a vector.
An example semantic problem occurs, when one needs to find a zero element or zero $\bar{0}$. In order to do this for sequences one'd intuitively define $\{a_n\}, a_n=0 \forall n$ as the zero element. However, this is a sequence, not a vector.
As to why the treating of sequences as vectors may not be intuitive:
What is the difference between operators, functions, sequences and vectors?
 A: You should carefully read the accepted answer of the question you linked to. That question was based on misconceptions, and it seems yours is too.
Consider the following passage from that answer:

If the terms of a sequence are elements of a field $F,$ then that sequence is an element of the vector space, over $F,$ of all sequences with terms in $F,$ so it is a 'vector' in that sense.

The comments on that answer indicate that the problem is pedagogical: the early levels of instruction in math (which is all most people ever see)
give only limited perspectives into the field of mathematics.
It is likely that someone will see only finite-dimensional vector spaces,
or at least that only the finite-dimensional vector spaces will be identified as vector spaces.
The more general, advanced viewpoint is that anything that satisfies the formal definition of a vector space is a vector space, and its members are vectors.
There are various ways to state a formal definition of vector space
(here is an example),
but they generally admit the same models of vector spaces,
for example sequences of elements of a field $F$ as described in the passage above.
This may not be intuitive at first. Part of an education in mathematics is gaining experience with the application of the definition of the term vector space
until applications like this become intuitive.
A: A similar confusion arises in Toplogy. Given any topology on a given set such as the real numbers, then the members of that topology are called "open" by definition of toplogical space. However, unlike open intervals in the usual topology of the real numbers, there is nothing "open", except the name, about the open sets of a topological space. It is simply a matter of terminology, not ontology.
In the same way, given any vector space, the elements of the space as called "vectors" by definition of vector space. There need not be any connection with what is usually called a vector in Euclidean spaces. The name "vector" here merely means that the object is a member of a particular set which is equipped with the operations of vector addition and scalar multiplication and the operations satisfy certain properties. It is entirely possible that the objects of the vector space may have different operations and names in another context. This does not contradict their status as vectors also.
