Below is the image of theorem on the Upcrossing Inequality of Durrett's Probability: Theory and Examples.
What I don't understand is the remark in the paragraph after the theorem's proof. Why, no matter how hard you try, you can't lose money betting on a submartingale?
You can get the pdf on the author's webpage.
Reading it in plain language, out of context, this assertion is clearly false. It is possible (in theory) to lose all $n$ of your first $n$ bets.
But if we accept that this statement is just an informal restatement of the inequality $E(K\cdot Y)_n \geq 0,$ then all it is saying is that you cannot devise a betting strategy so bad that you should expect to lose money with it. (This assumes the underlying game is favorable to you, so that you have a submartingale.)