- #1
SMA_01
- 218
- 0
Prove that the only subset of ℝ with the absolute value metric that are both open and closed are ℝ and ∅.
I know I'm supposed to prove by contradiction, but I'm having trouble:
Suppose there exists a clopen subset A of ℝ, where A≠ℝ, A≠∅. Let [x,y] be a closed interval in ℝ, where x is in A and y is in A' (complement of A). Now, let b=sup{z[itex]\in[/itex][x,y]|z[itex]\in[/itex]A}. Then I know b[itex]\in[/itex]A or b[itex]\in[/itex]A'.
I know that b is an upper bound for A implies b is a lower bound for A'. I'm just not sure how to arrive at a contradiction. I'm still not grasping the intuition behind it, can anyone explain intuitively what this means?
Thanks.
I know I'm supposed to prove by contradiction, but I'm having trouble:
Suppose there exists a clopen subset A of ℝ, where A≠ℝ, A≠∅. Let [x,y] be a closed interval in ℝ, where x is in A and y is in A' (complement of A). Now, let b=sup{z[itex]\in[/itex][x,y]|z[itex]\in[/itex]A}. Then I know b[itex]\in[/itex]A or b[itex]\in[/itex]A'.
I know that b is an upper bound for A implies b is a lower bound for A'. I'm just not sure how to arrive at a contradiction. I'm still not grasping the intuition behind it, can anyone explain intuitively what this means?
Thanks.