- #1

Zondrina

Homework Helper

- 2,065

- 136

## Homework Statement

Two parts to this question.

a) Suppose ##S## is a bounded set and has an interior point ##Q##, prove that ##C(S) > 0##. ( Hint : Inside any neighbourhood of ##Q##, you can place a

__square__).

b) For each integer ##n>0##, let ##L_n## be the line segment ##\{ (\frac{1}{n}, y) : 0 ≤ y ≤ \frac{1}{n} ) \}##.

Also, let ##S = \bigcup_{n=1}^{∞} L_n##, prove that ##C(S) = 0##.

## Homework Equations

##C(S)## = Outer content = ##inf \{ \sum A_i \} = inf \{ Area(P) \}##.

Where ##A_i## is the area of the sub-rectangle ##R_i##, which comes from a larger rectangle ##R## that encloses ##S## and has been partitioned by ##P##.

## The Attempt at a Solution

a) This part is not too bad I think.

Since ##Q## is an interior point of ##S##, ##\exists \delta > 0 \space | \space N_{\delta}(Q) \subseteq S##.

Now, place a square, with center ##Q##, inside of ##N_{\delta}(Q)## and let ##A_{S_Q} = Area(Square_Q)##.

Suppose that we now enclose ##S## in a rectangle ##R##. For any partition ##P## of ##R##, we can attain another partition ##P'## from the square, which yields the following relationship :

##inf \{ \sum A_i \} ≥ inf \{ \sum A_{i}^{'} \} ≥ A_{S_Q} > 0##.

Hence the result is shown, ##C(S) ≥ A_{S_Q} > 0##.

b) This part is a bit more difficult. I was thinking to take a square with area ##δ^2## that will contain all the line segments ##L_n## except a finite amount of them.

Last edited: