1. ## Supremum Question

Suppose A and B are non-empty sets that are both bounded above, and that for all X in A, there exists a Y in B such that x is less than or equal or y.

a) Prove that sup A is less than or equal to sup B (Need a general proof)
b) Do A = (6/7, 1) and B = {n/(n+1): n in N} satisfy the conditions? Explain.

2. Originally Posted by Janu42
Suppose A and B are non-empty sets that are both bounded above, and that for all X in A, there exists a Y in B such that x is less than or equal or y.
a) Prove that sup A is less than or equal to sup B (Need a general proof)
b) Do A = (6/7, 1) and B = {n/(n+1): n in N} satisfy the conditions? Explain.
We know that $\alpha = \sup (A)\,\& \,\beta = \sup (B)$ both of these exist.
Now suppose that $\beta < \alpha$. That means that $\beta$ is not an upper bound for $A$.
That means $\left( {\exists x \in A} \right)\left[ {\beta < x \leqslant \alpha } \right]$
But by the given $\left( {\exists y\in B} \right)\left[ {x \leqslant y} \right]$