It's intuitively obvious that this is true, and it's probably trivial to prove it mathematically, but I am several decades out of practice at writing proofs so I can't quite figure out how to prove it.
If you have a sample of numbers and all you know is its average, not the individual values or how many; and then you add a bunch more numbers to the sample, and the new numbers have an average higher than the original sample's average, the new cumulative average must be higher.
As a word problem: If a sports player has an average on some statistic for his career, then next year, his average for that year is higher than his average for his career up to that year, his new career average must have increased.
In mathematical lingo:
Let A and B be a set of numbers, with at least one element each.
Let mA and mB be the averages of the values in these sets.
Let C be the union of sets A and B; then mC is the average of the values of the set C.
If mB > mA, it follows that mC > mA.
Trying to write a rigorous mathematical proof, I get nowhere. I start with tA/nA < tB/nB and try to do things to that inequation (adding things to both sides, for instance) in hopes of getting closer to (tA+tB)/(nA+nB) > tA/nA but I never get anywhere.
I suppose there's some small chance that this is one of those things where proving it is a lot harder than it seems intuitively, but more likely, it just means that in the twenty years since my college days, I have forgotten almost everything. Meh.