Balancing Rice Bags

Brice has 6 bags of rice--labeled with their respective weights in pounds--which are arranged from left to right as follows: $1 \quad 3 \quad 5 \quad 4 \quad 7 \quad 6 .$ He wants to choose a collection of bags which adds up to exactly half the total weight of the rice. He proposes the following algorithm to solve the problem with his beam balance:

1. Put the first weight on the left pan, and the second weight on the right.
2. Pick up the next weight, and put it on the side of the balance with less weight (or any arbitrary side when the balance is balanced).
3. Repeat step 2 until all the weights are on the balancing beam.

As shown in the animation below, Brice has balanced out the beam successfully using the above algorithm. Does that mean that Birce can always use the same steps to balance it out regardless of the arrangement of the bags?

The algorithm in action