Assuming you have already explained the maths as others have suggested (70-60)+(90-80)=20 then here's a few other ideas...
Method 1 - switching the numbers up
A man buys a goat for £1 and then sells it for £1,000,000. He then buys another goat for £1,000,000 and sells it for £1,000,001. In this case the first sell and new buy price are deliberately the same. So in this example he uses his profit to buy the second goat but hopefully it is obvious to all that he started with £1 and ended with £1,000,001 and thus his profit is £1,000,000 and not just the £1 from the second sale.
You could then extend this to more akin to the original example with buying for £1 and selling for £1,000,000 and then buying for £1,000,001 and selling for £1,000,002. Again I hope it would be obvious that the seller has ended with a lot more than £1 profit.
Method 2 - Actually do it
Offer your friend an experiment. You both start with £100 (you can go for pennies if you prefer to be a bit less rich) and then a random item and perform the transactions as described.
A buys a goat from B for £60. A now has £40 and B has £160.
A sells the goat to B for £70. A now has £110 and B has £90.
A buys the goat from B for £80. A now has £30 and B has £170.
A sells the goat to B for £90. A now has £120 and B has £80.
Method 3 - separate stacks of money
Consider the first man to have two separate stacks of money, one with 60 in and one with 80 in. He uses these two stacks for his two purchases. Each stack will be £10 bigger than when he started, thus £20 profit. The separate stacks ensures that there is no possibility that you could consider the profits of the first set of transactions to have been consumed by the second set of transactions.
Method 4 - Reorder the transactions
A man buys two goats, one for 60 and one for 80, thus two for 140. He then sells them both for 70 and 90, or both for 160. Thus 20 profit. This technique means that the profit doesn't even exist at the time the second goat is purchased and thus there is no way that it could have been made to go away by the second goat purchase.
If this still fails to convince him then offer to actually perform these transactions til this other person has no money left and you have all their money. :)