My question is about the definition of cointegrated.

$y\_t =y\_{t-1}+u\_t$

$u\_t =\eta\_t +0.5\eta\_{t-1}$

where $\eta\_t\sim N(0,1)$ is i.i.d. white noise.

I claim that $y\_t$ and $y\_{t-1}$ are cointegrated because $y\_t -y\_{t-1}=u\_t$ is stationary and both $y\_t$ and $y\_{t-1}$ are non-stationary.

Is this a correct application of the definition? I know it would be more standard to just consider $y\_t$ as I(1), but if I said $y\_t$ and $y\_{t-1}$ are cointegrated, is that a mathematically correct statement?

EDIT: Firstly, thank you to everyone that helped.
The "attempt" fails as 1muflon1 describes.

EDIT2: Some people are currently not satisfied with the answers, and an ongoing discussion is [here](https://stats.stackexchange.com/questions/540337/is-a-random-walk-cointegrated-with-its-own-lag?noredirect=1#comment991993_540337).