You're so close!

You have a nice way of writing 4-u_{k+1} in terms of u_k. So now you need to show that , which in turn shows that 4-u_{k+1}<4. This should be easy for you, since you have assumed that u_k < 4. Use that to show that the above fraction is < 4, thus 4-u_{k+1}<4, thus u_{k+1}<4. This shows that u_{k} < 4 implies u_{k+1}<4, which is the proof.