TITLE: Proofs of theorems that proved more or deeper results than what was first supposed or stated as the corresponding theorem
QUESTION [54 upvotes]: Recently, I figured out that a colleague of mine has had published during recent years a proof of a theorem in which he was actually proving a deeper result which we both thought to be still open. After a closer look at his proof I found that, taking a bit more care and putting some additional emphasis in certain parts of his previous proof, he was actually proving the other still-thought-to-be-open problem: the construction was absolutely the same and therefore the proof of the previously published theorem was certainly a better argument than we first thought. I am curious now about this phenomenon happening more often. Do you know some other recent (let's say from 1700 to the current day) examples of this phenomenon of proofs being stronger than initially stated or proving more than thought at first?

REPLY [2 votes]: In the paper P. Erdős and A. Hajnal, On the structure of set-mappings, Acta Math. Acad. Sci. Hungar. 9 (1958), 111-131, the authors came close to solving Ulam's measure problem by proving that the first uncountable inaccessible cardinal is not measurable, as they pointed out in their subsequent paper P. Erdős and A. Hajnal, Some remarks concerning our paper "On the structure of set-mappings" — non-existence of a two-valued $\sigma$-measure for the first uncountable inaccessible cardinal, Acta Math. Hungar. 13 (1962), 223-226:

In accordance with the notations of [4] we say that a cardinal $m$ possesses property $P_3$ if every two-valued measure $\mu(X)$ defined on all subsets of a set $S$ of power $m$ vanishes identically, provided $\mu(\{x\})=0$ for every $x\in S$ and $\mu(X)$ is $m$-additive.
It was well known that $\aleph_0$ fails to possess property $P_3$ and that every cardinal $m\lt t_1$ possesses property $P_3$ where $t_1$ denotes the first uncountable inaccessible cardinal.
Recently A. Tarski has proved, using a result of P. Hanf, that a certain wide class of strongly inaccessible cardinals possesses property $P_3$ (called strongly incompact cardinals). H. J. Keisler gave a purely set-theoretical proof of this result. After having seen these papers we observed that the special case of this result that $t_1$ possesses property $P_3$ follows almost trivially from some of our theorems proved in [1]. We are going to give this simple proof in §2. Our method for the proof is of purely combinatorial character, and although it is certainly weaker than that of A. Tarski and H. J. Keisler, we think that it is of interest to formulate how far one can go with these methods at present.