Hmm. Well, according to your setup, you need to show that

for some

You either need to use inequalities in a rigorous fashion here, or pick some values for x and y for which you can compute various quantities exactly. Suppose x = 0, y = 1. Then you've got to show

or

I don't think those two values will work for you. However, you will notice that I did not use decimal approximations. I could try x = 4/5, and y = 1. Both are in the desired interval. Then I need to show that

or

or

Not sure how you can do that. The inequality holds, but I'm not sure how to prove it. Again, this is just an illustration so you can see how it's done.

Maybe it's easier, after all, just to show that the function's derivative is greater than 1 in magnitude somewhere on the interval (which it is).