Is the following argument correct?
Sequences $(x_n)$ and $(y_n)$, where $(x_n)$ converges, $(y_n)$ diverges, and $(x_n+y_n)$ converges.
Proof. The request in question is impossible. Assume that we have sequences $(x_n)$ and $(y_n)$ such that $(x_n)\to \alpha$, $(y_n)$ diverges and yet $(x_n+y_n)\to\beta$ where $x,\alpha\in\mathbf{R}$.
Then by combining the first two propositions of theorem $\textbf{2.3.3}$ we have $\lim(y_n) = \lim((x_n+y_n)-x_n) = \beta-\alpha$, contradicting our assumption that $(y_n)$ was not convergent.
$\blacksquare$
Note: The propositions in question are that Given $(a_n)\to a$ and $(b_n)\to b$ it follows that $\lim ca_n = ca,\forall c\in\mathbf{R}$ and $\lim(a_n+b_n) = \lim a_n+\lim b_n$