Okay so here are some questions I am having trouble with.
I did this proof and the professor marked it completely wrong, but I don't think I did anything wrong. If you could point out where I went astray that would be fantastic.
Question: Let , and . Show that x_n is a decreasing sequence.
We will show is a decreasing sequence using induction. Assume that for all . says which is true, so it holds for n=1. Assume that . This says that . So , which says . Thus . Therefore by induction, , which by definition says that is a decreasing seq.
And here are some general questions I am having trouble answering, so any hints are appreciated.
1)Assuming that the function f(x)= is continuous for all x, prove that there is an x (0,1) with .
, so the plan of attack I have for this proof is to let and prove that f(x)=g(x) for some x in the interval somehow. I am guessing it will involve using the definition of continuity somewhere along the line.
2) Prove that if f:[a,b] is continuous and f(x)>0 for x [a,b], then there is a constant k>0 so that for all x