Why does this MOSFET turn on initially in saturation?

[CoolDude420]

Homework Statement

Razavi's book (Analgoue CMOS Integrated Circuits) says that that "as Vin approaches Vth, M1 begins to turn on, drawing current from Rd and lowering Vout. Transistor M1 turns on in saturation regardless of the values of Vdd and Rd."

Why is that? I mean that the condition for saturation is (assuming Vgs > Vth) is:
For Saturation:
$$V_{DS} > V_{GS} - V_{TH} $$
Simplifies to:
$$V_{out}>V_{in} - V_{TH}$$

How does that prove it will turn on in saturation?

Homework Equations

The Attempt at a Solution

 

[Merlin3189]

Personally, I look at the characteristic curves rather than the formulae, but either way is ok.

When the mosfet is cut off, what is the output voltage?
And the input voltage?
Now, as the input voltage rises, at what level does the mosfet start to turn on?
What is the output voltage just before or just as the mosfet starts to turn on?
Are your inequalities satisfied?

Otherwise, on a I_D vs V_DS graph of characteristic curves, draw in a load line for R_D (any nominal value will do.) The output voltage will lie on the load line. Look at the point where it is just switching on. What region must it be in?