2. (a) Let B be a non-empty subset of real numbers that is bounded below with lower bound B E R. Prove that B = inf(B) if and only if there exists a sequence (bn) such that bn E B for each n E N and lim(bn) = B. (b) Let A be a non-empty subset of real numbers that is bounded above with a > -1 for all a E A. Define B to be the set B = 1 - : a E A Using part (a), or otherwise, prove that inf(B) exists and 1 inf (B) = 1+ sup( A) (c) Prove that sup(A) = 2/3 with 4n - 3 A = -:nENS. 6n (d) Using part (b), or otherwise, prove that the infimum of 6n B = 10n - 3 inEN exists and find its value explicitly. [19 markelIn part (b), we want to determine the infimum (greatest lower bound) of the set B, defined as; B = lita :aga. First, we need to establish the existence and value of inf(B). We know that A is a non-empty subset of real numbers that is bounded above by -1 because we're given that -1 is an upper bound for A. Therefore, we can conclude that sup(A ) exists and is equal to -1 Now, we want to find inf(B). According to the properties of infimum, if we can find a sequence (bn) such that bn @ B for each n @ N (meaning the elements of this sequence are not in B) and lim(bn) = inf(B), we can determine inf(B).Consider the sequence (an) defined as follows: This sequence consists of values that are greater than sup(A) = -1 for each n, and an @ A for each n because an is not an element of A. Now, let's define the corresponding sequence (bn) as: bn = 1+an Now, let's address your question about the limit of the sequence (bn). You mentioned that you expected lim(bn) to be equal to a symbol "beta" (8), and it seems like you might be referring to a different result. In this specific problem, we are not trying to find a constant "beta," but rather, we want to find inf(B), which is the greatest lower bound of the set B.In the context of this problem, we want to show that inf(B) exists and find its value. To do that, we need to show that there exists a sequence (bn) such that bn