Is the following statement true for all functions f and g?

[frootloopers]

∃x ∈ R s.t. ∀y ∈ R, x < y ⇒ |f(x) − g(x)| < |f(y) − g(y)|

Hi, I wanted to confirm this since it's for a homework assignment; I personally think that the above statement would always be true, since there is no real number where all real numbers are less than it. Thus, the first condition of the implication is false, and the implication will return a true.

 

[ksdhart2]

You seem to misunderstanding the notation. It's saying if the first condition is true, then the second condition is also true. It's not saying that the first condition must be true (and, in fact, you're absolutely correct that it's false most of the time).

Rather, the problem asks you to pick some arbitrary \(x\) and then consider all real numbers \(y\) that are strictly less than this arbitrary value, and see if it's then true that \(|f(x) − g(x)| < |f(y) − g(y)|\) for any functions \(f(x)\) and \(g(x)\).

Remember that in order to prove a statement false you only need one counter-example. Suppose \(f(x) = x\) and \(g(x) = x^2\). What do you find then?

 

[Steven G]

Suppose g(x)=0. Now consider ksdhart2 example with f(x) = x²