A simple theorem on the utility function

Claim:

For a two-good system, if the utility function is of the simple
form:

U=U1+U2

then neither 1 nor 2 can be inferior.

Proof:

(a) lemma: in a two-good system, there are only three possible
combination (N,N), (N,I), (I,N) where "N" stands for normal and "I"
for inferior. So two goods cannot be both inferior. I won't be
bothered to prove this lemma and take it for granted. (Beat me?)

We also need the following two principles.

(b) First, we know "the principle of maximizing utility" is
equivalent to "equalizing marginal utility per dollar", and then is
equivalent to "MRS=relative price" where MRS stands for marginal
rate of substitute.

(c) Second, "diminishing marginal utility" implies for Q
(dU/dq)(Q)>(dU/dq)(Q'), namely the marginal utility is a monotonic
decreasing function in quantity q.

Alright, reductio ad absurdum:

Assume good 1 is inferior. We test this claim by increasing the real
income. By (a), good 2 has to be normal. Let the original quantities
at which utilities are maximized be Q1, Q2. Income effect tells us
at the new tangent point with maximized utility, if the quantities
are Q1', Q2' then

Q1>Q1'

and

Q2'>Q2.

Then based on (c), we conclude

(dU1/dq1)(Q1')>(dU1/dq1)(Q1) and

(dU2/dq2)(Q2')<(dU2/dq2)(Q2).

Therefore, MRS for these two cases cannot be equal.

However, the relative price is the same for these two cases. So
based on (b), we arrive at a contradiction. The claim follows.

QED

Remark:

If one good is inferior in a two-good system then the utility
function must assume an involved correlated form.