The contradiction of the equation of motion of a free electron

I’ve been confused of the equation of motion of a free electron inside a conductor as a current flows.

$mathbf{v}:=$average velocity of free electrons inside the conductor.

$tau:=$relaxation time.

$m:=$mass of a free electron.

$-mfrac{mathbf{v}}{tau}:=$resistance which acts against the free electron.

$mfrac{d(mathbf{v})}{dt}=e*mathbf{E}+left(-mfrac{mathbf{v}}{tau}right)$

I’ve written the below my assumptions.

$mfrac{d(|mathbf{v}|)}{dt}=|emathbf{E}|+left(-mfrac{|mathbf{v}|}{tau}right)$

$0leq mfrac{d(|mathbf{v}|)}{dt}$ is non decreasing.

$0leq |emathbf{E}|$ is constant.

$0leq|mathbf{v}|$ is non decreasing and the value of it finally takes some constant value.

$0geqleft(-mfrac{|mathbf{v}|}{tau}right)$ is non increasing and finally it will take some constant value.

Of course the contradiction arises as above my assumptions are applied.The left term of the equation is non decreasing so the right side of the equation must also to be non decreasing however my assumptions hinder it.

Can anyone tell my what I’ve missing or the site which describes of above things? Or is above equation of motion is incorrect?