TITLE: Examples of open problems solved through short proof
QUESTION [11 upvotes]: Are there good examples of reasonable open problems in mathematics that had an 'obvious' solution via application of a theorem already known/not yet found in mathematics but could have been found with some effort but went unnoticed in mathematics until someone provided a footnote with the 'obvious' solution? Proof needs to be reasonably short than that possibly expected.
Euler bridge problem qualifies however I am looking at something that would be published post 1900s or preferably 1950s in mathematics.

REPLY [9 votes]: a semi famous example from number theory, Erdos Proof of Bertrands postulate (paper by Galvin). 

In 1845 Bertrand postulated that there is always a prime between n and 2n, and
  he verified this for $n < 3 \cdot 10^6$
  . Tchebychev gave an analytic proof of the postulate in
  1850. In 1932, in his first paper, Erdős gave a beautiful elementary proof using nothing
  more than a few easily verified facts about the middle binomial coefficient. We describe
  Erdős’s proof and make a few additional comments, including a discussion of how the
  two main lemmas used in the proof very quickly give an approximate prime number
  theorem. We also describe a result of Greenfield and Greenfield that links Bertrand’s
  postulate to the statement that $\{1, . . . , 2n\}$ can always be decomposed into $n$ pairs such
  that the sum of each pair is a prime.