Let s ⊆ r be nonempty. prove that if a number u in r has the properties: (i) for every n ∈ n the number u − 1/n is not an upper bound of s, and (ii) for every number n ∈ n the number u + 1/n is an upper bound of s, then u = sup s.

Paula makes stained-glass windows and sells them to boutique stores. if her costs total $12,000 per year plus $4 per window for the frame. how many windows must she produce to earn a profit of at least $48,000 in one year if she sells the windows for $28 each? 1. define a variable for the situation. 2. write an inequality that represents her profit. note: revenue is money coming in. cost is money going out. profit is the difference between the revenue and the cost. in other words: revenue - costs profit 3.using words, describe how many windows she must sell to have a profit of at least $48,000.