I am having some difficulty in understanding this problem. 1. Prove that there is a real x, s.t x-|_ x _|=2 or that there isn't. (|_ x _|= m (element of) Z and m<=x<m+1.) Read as, "the floor function of x is equal to m, element of Z(integers), and m is less than or equal to x, which is less than m+1. How am i suppose to solve this problem, and what kind of definitions or facts can i use to solve this problem? Could someone tell me the steps that are required to solve this problem? And this is the second problem. 2. Show that there is a real x s.t. f(x)=x^5+x+1=0. f(x) is continuous because it is a polynomial. Let s be a real s.t. f(-s)<0<f(s). Then by I.V.T.(intermediate value theorem) there is at least one x in (0-s,0+s) s.t. f(x)=0 if f(-s)<0<f(s). Could someone just clarify this for me, because i think i am able to do it, just dont understand on what to do first. Thanks in advance.