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 $\displaystyle x_1=a \geq 2$, and $\displaystyle x_{n+1}=1/4(x_n + 5) $. Show that x_n is a decreasing sequence.

My proof:

We will show $\displaystyle \{x_n\} $ is a decreasing sequence using induction. Assume that $\displaystyle x_k \geq x_{k+1} $ for all $\displaystyle k \leq n $. $\displaystyle x_1>x_2 $ says $\displaystyle 2>\frac{7}/{4} $ which is true, so it holds for n=1. Assume that $\displaystyle x_k \geq x_{k+1} $. This says that $\displaystyle 4x_{k+1}-5 \geq x_{k+1} $. So $\displaystyle 4x_{k+1} \geq x_{k+1}+5$, which says $\displaystyle x_{k+1} \geq \frac{1}{4}(x_{k+1}+5) $. Thus $\displaystyle x_{k+1} \geq x_{k+2} $. Therefore by induction, $\displaystyle x_n \geq x_{n+1} $, which by definition says that $\displaystyle \{x_n\} $ 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)= $\displaystyle 2^x $ is continuous for all x, prove that there is an x $\displaystyle \in $ (0,1) with $\displaystyle x2^x=1 $.

$\displaystyle x2^x=1 says 2^x =\frac{1}{x} $, so the plan of attack I have for this proof is to let $\displaystyle g(x)=\frac{1}{x} $ 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] $\displaystyle \rightarrow \Re$ is continuous and f(x)>0 for x $\displaystyle \in$ [a,b], then there is a constant k>0 so that $\displaystyle f(x) \geq k$ for all x $\displaystyle \in [a,b].$