Seq of real numbers

Question

Let the seq ${X(n)}$, ${Y(n)}$ are divergent. Then

1. The seq ${X(n)+Y(n)}$ may or may not be convergent.
2. The seq ${X(n)+Y(n)}$ always divergent.
3. The seq ${X(n) •Y(n)}$ may or may not be convergent.
4. The seq ${X(n)•Y(n)}$ is always divergent.

Answer 1

$X_n=0$ for odd $n$ and 1 for even $n$, whilst $Y_n=1$ for odd $n$, 0 for even $n$. Then $X_n+Y_n$ diverges and $X_n\cdot Y_n$ converges.

If $X_n$ is as above, but $Y_n=X_n$, then $X_n+Y_n$ diverges and $X_n\cdot Y_n$ diverges.

If $X_n$ is as above, but $Y_n=-X_n$, then $X_n+Y_n$ converges and $X_n\cdot Y_n$ diverges.

So 1. and 3. are true.