The boundness of sequence $\{y^{k+\frac{1}{2}}\}$

Let sequence $\{y^{k+\frac{1}{2}}\}$ generated by$$y^{k+\frac{1}{2}}=y^{k}+z^{k},$$where $z^{k}$ not be necessary bounded. If $\{y^{k+\frac{1}{2}}\}$ is bounded, can we deduce$\{y^{k}\}$ is bounded?

Here is my consideration, is it right?

If we take $y^{k+\frac{1}{2}}=\frac{1}{2}$, $y^{k}=n+\frac{1}{4}$ and $z^{k}=\frac{1}{4}-n$, then$\{y^{k}\}$ is unbounded. If $\{z^{k}\}$ is bounded, then $\{y^{k}\}$ is must bounded.