I've been reading Tomas Bjork's 'Arbitrage theory' and it says:
To say that $d ≤ (1 + R) ≤u$ holds is equivalent to saying that $1 + R$ is a convex combination of u and d, i.e. $1 + R = q_u · u + q_d · d $
I understand why the condition $d ≤ (1 + R) ≤u$ should hold for there not to be an arbitrage opportunity, and I also understand that $1 + R = q_u · u + q_d · d $ means that the expected return of the stock is equal to the risk-free return, but how does that inequality holding imply this equality? What's the proof to get to this convex combination?