Let Show that is bounded above, then

Okay, so I first showed that is bounded below, that's easy, but as for second part, I want to know whether my work is correct or not:

Since is bounded above, then it has supremum, so (from this and the rest, sup and inf are taken for ), and on the other hand since and are bounded above and below, respectively, then it exists a so that and namely we have and hence

If it's not correct, what's the way to do this?