1. ## Proving Theorems

Hello,

Alternatively there are more theorems in the book Further Mathematics for economics analysis, page 54, chapter 2 Theorem 2.4.2

There is a little explanation on the theory but it doesn't explain too much detail.

Thanks

2. ## Re: Proving Theorems

Okay this is what I have for theorem b of the book:

I'm using the explanation for theorem a) in the book (at the bottom of page 54)

f is convex, the set $\displaystyle p^a$ = $\displaystyle x \forall S : f(x) <= a$ is convex for every number a

I'm not fully understanding what I'm doing here but here is my answer;

let x and y be points in $\displaystyle P^a$
Then x and y belong to S

NOTE: [Am i right in saying that x is the f(x) = f(xa...xn) and y is the g(x) = g(x1....xn) in this proof?]

Continuing:

f(x) <= a
f(y) <= a

IF λ $\displaystyle \forall$ [0,1]

then $\displaystyle λx + (1-λ)y$ belongs to S

Since S is a convex set...

As f is convex also; (I move now to the definition on page 54 of a convex set)

$\displaystyle \lambda x + (1- \lambda) y$ >= $\displaystyle \lambda f(x) + (1- \lambda) f(y)$

for all x and y in S and all Lambda in [0,1]

Shows that $\displaystyle \lambda x + (1- \lambda) y$ in all $\displaystyle p^a$

confirms $\displaystyle p^a$ is convex.....

Is this correct?

Honestly I just changed the words from the proof of part a on page 54, and don't really understand the Theorem

Any help would be much appreciated!