Sequences — when the $\liminf$/$\limsup$ chain is not strict

For any two sequences $\{x_n\}$ and $\{y_n\}$ it is known that

$$\liminf x_n+\liminf y_n \le \liminf (x_n+y_n) \le \limsup (x_n+y_n) \le \limsup x_n+\limsup y_n.$$

Suppose $x_n=0$ for $n$ even and $x_n=1$ for $n$ odd, and $y_n=1-x_n$ for all $n\ge1$. Which statements are true?

$S_1$: strict inequalities hold in $(*)$ above.
$S_2$: $\{x_n+y_n\}$ converges, while $\{x_n\}$ and $\{y_n\}$ do not.