$($a$)$ Find the upper and lower limits$,$ if exist, of $\{\frac{2x^2}{7}-[\frac{2x^2}{7}]\},$ where $x$ is the greatest integer $x.$ Is the sequence convergent? Justify.

$($b$)$ Prove or disprove : For any two bounded sequences $\{x_n\}$ and $\{y_n\}$ of real numbers, $\underset{x_n+y_n}{lim}=\underset{x}{lim}{?}_n\frac{+}{b}ar(\underset{?}{lim})y_n.$

$(2+1)+2$