You can simplify the mathematical argument a lot by looking at DCA as forcing a relative "buy low sell high." When the price of the asset is lower, your DCA buys more, when it's higher you buy less. Relative to a single purchase at the average price, this is equivalent to buying low and selling high.
Imagine you commit to buying $100 on each of date A, B, and C. On A, the price is $10/share. You buy 10 shares. On B, it’s $100/sh. You buy 1 share. On C, it’s $1/sh. You buy 100 shares. You have 111 shares at a cost-basis of $300 (or $2.70/share).
The time-weighted average price was $37/share.
DCA is a way to force yourself to make purchases at a variety of prices and times. (How likely is it with a single-purchase strategy that you'd own 111 shares at $2.70/sh of a stock that charted $10->$100->$1? How would you feel when you saw your $300 investment go to $111 in value? DCA takes a lot of that psychological pressure off and is amenable to decide-once and automate.)
