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tarheelborn
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If [tex]S[/tex] is a connected subset of [tex]\mathbb{R} [/tex] and [tex]S[/tex] is bounded below, but not above, then either [tex]S=[a, \infty)[/tex] or [tex]S=(a, \infty)[/tex] for some [tex]a \in \math{R}[/tex].
tarheelborn said:Yes I am familiar with the types of connected subsets of [tex] \mathbb{R}[/tex]. I know that if [tex]a<b \in S[/tex] then [tex] [a,b] \subseteq S[/tex].
Being bounded below means that there exists a real number, called the lower bound, that is less than or equal to all the numbers in the subset. In other words, the subset has a finite or infinite minimum value.
To determine if a subset of real numbers is bounded below, you can compare the numbers in the subset and see if there is a minimum value. Alternatively, you can use the concept of infimum, which is the greatest lower bound of a subset, to determine if a subset is bounded below. If the infimum exists and is a real number, then the subset is bounded below.
A subset of real numbers is bounded below if it has a finite or infinite minimum value, while a subset is bounded above if it has a finite or infinite maximum value. In other words, a subset that is bounded below has a lower bound, while a subset that is bounded above has an upper bound.
Yes, a subset of real numbers can be both bounded below and unbounded above. This means that there exists a minimum value for the subset, but there is no maximum value. For example, the set of all negative real numbers is bounded below by 0, but it is unbounded above because there is no maximum negative number.
No, a subset of real numbers can be bounded below and unbounded above and still be a finite set. For example, the set of real numbers between 0 and 1, including 0 and 1, is bounded below by 0 but is unbounded above because there is no maximum value. However, this set is finite because it only contains a finite number of values between 0 and 1.