Determine if each of the following sets S is bounded above...

Determine if each of the following sets S is bounded above or below. Find each of the following (if it exists) max S, min S, sup S, and inf S. If a value does not exist write DNE. a) S = (2,7) \cup [8,12] b) S = [-4,2] \cap (-1,\infty) c) S = \{2 \cdot (-1)^{n} + \frac{(-1)^{n+1}}{n} | n \in \mathbb{N}\} d) \bigcup_{n=1}^{8} [\frac{1}{n}, n]

Transcript

00:01 Okay, so in this question we're asked to consider these three sets a, b and c and find the inf, the sup, the min and the max.

00:10 Okay, if inf and sup always exist, might be infinite, then min and max not always exist.

00:16 So min and max will be the inf and the sup when they belong to the set.

00:20 So let's start with set a and we know the value of what set a does, right? so set a starts at zero and then is one half and then is two thirds and three over four, all the way and the limit is one as n goes to infinity, right? so one first observation is that the inf is the smallest number.

00:45 So the inf of a coincides with the min of a, which is zero, right? zero belongs to the set.

00:52 Now the sup of a, we also know what it is.

00:57 The sup will be the limit of this sequence and it is one.

01:01 But because one does not belong to the set, there is no maximum element, max of a, right? maximum a would be one, but one does not belong to the set, right? we are never one.

01:14 We get infinitely closer to one, but we are never one.

01:17 So the max does not exist.

01:20 Let's look at part b now.

01:22 So part b, let's look at the positive numbers in part b and what we have there is one over two, three over four.

01:31 Then we have, maybe i skipped the, did i skip the seven? okay, seven over eight.

01:36 We are going in powers of two, right? then we have 15 over 16, et cetera.

01:43 And this is going to one.

01:45 And on the negative side, we have minus one over two, minus three over four, minus seven over eight, minus 15 over 16.

01:52 And this is going down to minus one, okay? and here i can even say this is going up to one...

Question

Please give Ace some feedback