22/7 - the closest mathematical approximation of Pi

Proofs of the mathematical result that the rational number 

22/7 is greater than π (pi) date back to antiquity. One of these proofs, more recently developed but requiring only elementary techniques from calculus, has attracted attention in modern mathematics due to its mathematical elegance and its connections to the theory of Diophantine approximations. Stephen Lucas calls this proof "one of the more beautiful results related to approximating π".[1] Julian Havil ends a discussion of continued fraction approximations of π with the result, describing it as "impossible to resist mentioning" in that context.[2]

The purpose of the proof is not primarily to convince its readers that 22/7 (or 3+1/7) is indeed bigger than π; systematic methods of computing the value of π exist. If one knows that π is approximately 3.14159, then it trivially follows that π < 22/7, which is approximately 3.142857. But it takes much less work to show that π < 22/7 by the method used in this proof than to show that π is approximately 3.14159.