Final answer:
The supremum of the set is 1 and the infimum is 1/2. The sequence is bounded above by 1 and bounded below by 1/2.
Step-by-step explanation:
To find the supremum and infimum of the set { 1 + (-1)ⁿ/n : n∈N }, we need to understand the behavior of the sequence as n increases. Let's first observe the terms of the sequence:
n=1: 1 + (-1)ⁿ/n = 1
n=2: 1 + (-1)ⁿ/n = 1 - 1/2 = 1/2
n=3: 1 + (-1)ⁿ/n = 1 + 1/3 = 4/3
n=4: 1 + (-1)ⁿ/n = 1 - 1/4 = 3/4
We can see that the terms alternate between 1 and a sequence that approaches 1. Therefore, the sequence is bounded above by 1 and bounded below by the infimum of 1/2. So, the supremum of the set is 1 and the infimum is 1/2.
Supremum: 1
Infimum: 1/2
Proof:
1. To prove that 1 is the supremum, we need to show that no number greater than 1 can be an upper bound of the set. Since the terms of the sequence approach 1 as n increases, any number greater than 1 will eventually be exceeded by a term in the sequence. Therefore, 1 is the supremum.
2. To prove that 1/2 is the infimum, we need to show that no number smaller than 1/2 can be a lower bound of the set. Since the terms of the sequence alternate between 1 and a sequence that approaches 1, no number smaller than 1/2 will be smaller than all the terms of the sequence. Therefore, 1/2 is the infimum.