I am working on the following exercise and could use some help.

Prove that if

, then

is integrable on

and

.[Hint: Any lower sum is

and some lower sum

. Hence, the lower integral

].

Here is what I have tried so far...

Let

be a lower step function such that

for all

. Then, let the lower sum for

. Then, we have that

for any lower step function. Also,

. Thus, for any

,

. So,

.

So for

, we see that any lower sum

and some lower sum is

. Thus,

=lower integral

.

We can do something similar for

...For

we have that

and

. Therefore,

.

Then conclude that

.

I'm not sure if I have said enough for the proof to be complete or if what I have is even good. Please help me out.