If f is integrable then where and .
You also know that right?
Okay so this makes sense because if the integral of f exists then kf should exist if k is an element in the reals.
okay so there is a theorem in my book that says:
Let a,b, and k be real numbers. a< b. Let f and g be a real-valued functions that are Riemann integrable on [a,b]. Then integral b to a of kf equals k* integral b to a of f.
So i start out by saying e>0 and that there is a delta>0 such that ||P||<delta.
||P|| = max(xi - x(i-1))<delta
I just can't figure out how to incorporate a real number in when I have been solving proofs like Prove (f+g) = (f)+(g) when f and g are both functions and greater than or equal to zero. When i can solve by using inf and sup to prove ...... can i do that here as well?
Any help would be appreciated .......
I will continue looking at the problem
1. (the same for infimums). Prove this.
2. From this we see that
3. Using this we see that . (think the definition of integrability).
This proves integrability.
Now, we know that .
Now apply the ideas of 1. to finish.