I'm learning about the exterior derivative from Loring Tu's An Introduction to Manifolds.
On pg. 38, the exterior derivative for a $k$-form $w$ is defined as:
For $k\geq1$, if $w=\sum_I a_Idx^I$, then $$dw=\sum_I da_I \wedge dx^I=\sum_I(\sum_j\frac{\partial a_I}{\partial x^j}dx^j)\wedge dx^I$$ where $I$ is the multi-index notation $(i_1,...,i_k)$ and $1\leq i_1 <...<i_k\leq n$.
But on pg. 40, while trying to prove $d^2=0$, the following step was written: $$d^2(fdx^I)=d(\sum \frac{\partial f}{\partial x^i} dx^i\wedge dx^I)=\sum \frac{\partial^2f}{\partial x^j \partial x^i}dx^j\wedge dx^i \wedge dx^J$$ where $f$ is a $C^\infty$ function.
I don't understand how the second exterior derivative is performed. To use the definition of the exterior derivative, we need to make sure that $\sum \frac{\partial f}{\partial x^i} dx^i\wedge dx^I$ is in the form $\sum_I a_Idx^I$, where the multi-index $I$ is in ascending order. How can we be sure that $dx^i\wedge dx^I$ when combined has the index in ascending order, i.e. $i <I$?