I think my text defines it as starting from 0. In that case, sigma is {(0,0),(1,1)}. Then it is a prefix of tau according to your example. Am I wrong?
This is a summary of a definition of a sequence from my text:
A sequence of n elements taken
from set A is a function mapping from {0,1,2,...,n-1} to A. If
we call the n elements a_1,a_2,...,a_n, then the sequence is the
function that maps 0 to a_1, 1 to a_2, 2 to a_3, ..., and n-1 to
a_n. Such a function, seen as a relation, is the set of ordered
pairs {(0,a_1),(1,a_2),...,(n-1,a_n)}. E.g., the sequence <
H,E,L,L,O >, is the set {(0,H),(1,E),(2,L),(3,L),(4,O)}.
Does this change anything that you were saying?
With this in mind I still can't come up with a proper counter example, nor can I understand your example properly.
I truly mean you no disrespect by this comment.
But this is an English language forum.
As such, we have very clear definitions of terms.
A sequence is a function from the positive integers to a field, real or complex.
If the sequence $\displaystyle \sigma_n$ is a subsequence of $\displaystyle \tau_n$ then $\displaystyle \sigma_n=\tau_{n_j}$ where $\displaystyle n_1<n_2<\cdots<n_n$.
In that westernized context, can you reframe your question?
If not, you need to find a forum that is compatible your language.
So technically with that definition of sequence, there is no counter example to show that sigma is a prefix of tau but not a subset. But if sequences don't have to be defined on an initial segment of natural numbers, then your example is a valid counter example.
Is this right?
Yes.
Edit: In case sequences don't have to be defined on an initial segment, my example shows that being a subset does not imply being a prefix. For the other direction under the same definition, let sigma = {(1,0)} and tau = {(0,0), (1,1)}. Then sigma (representing the sequence <0>) is a prefix of tau (representing <0,1>), but is not a subset of tau.