We can rewrite the given expression as:

1/2^2 - 1/2^4 + 1/2^6 - ... + 1/2^2002 - 1/2^2004
= (1/2^2 - 1/2^4) + (1/2^6 - 1/2^8) + ... + (1/2^2002 - 1/2^2004)

Notice that each pair of terms in parentheses can be simplified using the formula for the sum of a geometric series:

a(1 - r^n)/(1 - r)

where a is the first term, r is the common ratio, and n is the number of terms.

For example, the first pair of terms can be simplified as:

1/2^2 - 1/2^4 = (1/4)(1 - 1/4) = 3/16

Using this formula for each pair of terms, we get:

3/16 + 15/1024 + 255/1048576 + ... + (1/2^1998)(1 - 1/4) + (1/2^2000)(1 - 1/4)

Notice that the common ratio in each pair of terms is 1/4. Therefore, we can rewrite the entire expression as a single geometric series:

3/16 + 15/1024 + 255/1048576 + ... + (1/2^1998)(1 - 1/4) + (1/2^2000)(1 - 1/4)
= (3/16)(1 - (1/4)^500) / (1 - 1/4)
= (3/16)(4/3 - (1/4)^500)

Simplifying this expression, we get:

(3/4)(1 - (1/4)^500)

Now, we need to show that this expression is less than 1/2:

(3/4)(1 - (1/4)^500) < 1/2

Multiplying both sides by 4/3, we get:

1 - (1/4)^500 < 2/3

Adding (1/4)^500 to both sides, we get:

1 < 2/3 + (1/4)^500

Since (1/4)^500 is a very small number (approaching zero), we can conclude that the inequality is true and the given expression is less than 1/2.