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 .