a question about lebesque numbers

so we have that {U_alpha} where alpha is in A (indexing set) is a finite open cover of a compact metric space X. i did the first part of problem which was to show that there exists ε>0 such that for each x in X, the open ball B(x;ε) is contained in one of the U_alpha's.

so here such an ε is called the lebesque number of the cover (definition)

now i need to show that if at least one of the U_alpha's is a proper subset of X, then there is a largest lebesque number for the cover.

the hint given is that if one of the open sets in the cover is proper, the lebesque numbers are bounded. and since cover is finite, the least upper bound of lebesque number is again a lebesque number.

can someone explain the problem? thanks.