Medium run conclusions from Okun's Law and Expectations-Augmented Philips Curve
At least in the 3d edition, Blanchard writes:


  "In the medium run, the unemployment rate must be constant (my emphasis).
  The unemployment rate cannot be decreasing or increasing forever."


This is an inherent characteristic of the definition of the "medium run": that in it, some core magnitudes are constant. In some models, levels are constant (eg. aggregate consumption, aggregate capital stock, etc). In other models, growth rates are constant, and the model "has a steady-sate in growth rates". This means that levels grow indefinitely of course, but it is not wise (or useful, or meaningful) to project economic concepts to "infinite future". 

Now for unemployment, it makes no sense to not be constant in equilibrium (at this simple level of exposition). In more sophisticated models, one could have the unemployment rate approaching asymptotically a constant (zero or not) without ever actually reaching it, but this is usually mathematical gadgetry with not much of economic intuition. (And in futuristic models, one could have unemployment approaching asymptotically unity, as machines and Artificial Intelligence take over all jobs). 

Can this constant-by-conception-in-the-medium-run unemployment rate $u^* =const.$ be something else than the "natural rate of unemployment" $u_n$?

The natural rate of unemployment is  defined through the Phillips Curve, as that rate of unemployment at which realized inflation equals expected inflation (see previous chapters). Note that it is not assumed a priori that the natural rate of unemployment will be strictly positive -it could very well be zero. It is actual data that tells us that inflation stabilizes at strictly positive rates of unemployment. Then it is constant when inflation is a constant, and Blanchard discusses why inflation will also be a constant in the medium run. So it obtains that in the medium run $u_t = u_n =const.$.  

Since the current unemployment rate stabilizes when inflation becomes constant, which happens in the medium run, the constants must coincide. If $u^* \neq u_n$ inflation will not be constant and we would violate another condition for the characterization of the medium run.

And Blanchard writes correctly


  "In the medium run the unemployment rate must be equal to the
  natural rate of unemployment". 


So it does not assume what he wants to prove. The natural rate of unemployment is defined separately and with its own logic to be a constant, from the medium-run unemployment rate which is constant based on other arguments than those related to the natural rate of unemployment. 

I hope this helps.