I've recently started studying math finance from Shreve's Stochastic calculus text. In the binomial model, there is no arbitrage $\iff d<1+r<u$. To show that no arbitrage implies $1+r<u$, suppose $1+r\geq u$. Short sell $x$ stock and invest in the money market, so after time $1$, $x(1+r)\geq u>d$.
Again, I'm new to these financial concepts, but my understanding is at time zero, you assume you have zero wealth. So you'd have to borrow $x$ stock, but then wouldn't there be some type of interest that you would incur?