Negation of a statement about convergence How one can negate this statement:

A real sequence converges to a limit $x$ if and only if every subsequence also converges to the limit $x$.
I see no better negation than 

"It is false that a real sequence converges to a limit $x$ if and only if every subsequence also converges to the limit $x$."

A logically equivalent one is

"A real sequence converges to a limit $x$ and some subsequence does not converge to the limit $x$, or, a real sequence does not converges to a limit $x$ and every subsequence converges to the limit $x$."

Both are equivalent to false.
We can negate as follows

$$\lnot(A\iff B) \equiv A\iff \lnot B$$

that is

a real sequence converge to a limit x if and only if exists a subsequence which doesn't converge to the limit x