About the converse of below theorem

My book states that

if a sequence $a_n$converges to $l$, then every subsequence also converges to $l$.

Now it states at the end that its converse is not true. but as we know that if every subsequence converges to same limit $l$ then $a_n$ also converges to $l$. Kindly tell me if am wrong in understanding the converse or the author has made any mistake.There should be iff condition in the above theorem.