While reading some of proofs in Computability Theory, I came up with following conclusion:
We can design a Turing Machine which exclusively accepts finite strings (obvious).
Now while trying the same for infinite string, I came up with the following conclusion :
There exist no Turing Machine which exclusively accepts infinite strings.
Proof : Since acceptance in context to Turing Machine is defined in terms of finite amount of time, therefore every time we say that an infinite string $w$ is accepted by a Turing Machine, we usually talk about searching a "finite pattern" in $w$. So while designing a machine for $w$ there exist a $w'$ which is finite and contains the same "finite pattern" which we try to find in $w$.
Example of "finite pattern" : A string that contains a 0. (Assuming Binary Strings)
So my question is that is my proof right? and if yes, is there a better way to prove this?