Let's look at our two definitions, shall we?
Promise -- means to assure someone that one will definitely do, give, or arrange something; undertake or declare that something will happen.
Prove -- means to demonstrate the truth or existence of (something) by evidence or argument.
So, what they are saying is that they cannot promise you a good night's sleep but that they can prove you should/might be getting a good night's sleep with their mattress.
Don't you think the lawyers had something to do with how this commercial was worded?
Marketing/Advertising is all about getting the consumer to buy something that they do not really need while at the same time, getting them to pay more for that item than they should or can afford by using a credit card.
Another aspect of Marketing/Advertising falls on the design department and the production department along with the philosophy of UPPER MANAGEMENT by making sure we increase sales through REPEAT BUSINESS. Therefore, most products built in the good ole US of A are designed and manufactured to be replaced in 3-5 years.
Our lack of due diligence is allowing our companies to increase their profits substantially each year making it a good investment for their shareholders if they are a publicly traded company.
