For notation sake say .
Then we have:
If is a convergent sequence and . Show that for some .
My attempt was like this.
I used the limit limitation theorem*.
Assume for sufficiently large
Then,
A contradiction.
Thus, the statement for sufficiently large is false.
My difficutly is that we cannot conclude that .
Because that cannot be the negation of the statement.
It may mean that, there is no such such
for .
Meaning we can have,
for .
for .
for .
And thus on....
*)If is a convergent sequence and for sufficiently large then .