Suppose that S is a nonempty bounded set of real numbers and T is a nonempty subset of S.
(a) Show that T is bounded:
(b) Show that glb S $\leq$ glb T $\leq$ lub T $ \leq$ lubS
for (a) it seems intuitive that T is bounded because it exists in the bounded set S. But explaining this with some rigor is tough...
for (b) it seems also intuitive that since T is a subset of S it by necessity will have an equal or greater glb and an equal or lesser lub than S... but I'm struggling to explain it definitively... any thoughts?