The numbers $n+1,n+2,...,2n$ are written on a blackboard $(n ≥ 2)$, and the following procedure is repeated: two numbers are selected ( $x$ and $y$ ) from the board, erased, and replaced with the numbers $x+y+\sqrt{x^2 +y^2}$ and $x+y-\sqrt{x^2 +y^2}$ . Prove that there will never be a number less than 1.442 written on the board.
I've managed to find out that 1.442 is approximately $1/\ln(2)$ and that $1/(n+1) + 1/(n+2) + ... + 1/(2n) = 1 -1/2 + 1/3 -...-1/(2n) = \ln(2)$, which I feel should be really useful, but I have absolutely no idea how to utilize it.
$$\frac1{x+y+\sqrt{x^2+y^2}}+\frac1{x+y-\sqrt{x^2+y^2}}=\frac1x+\frac1y $$ Thus it remains to show that your $\approx \ln 2$ can be turned into a $<\frac1{1.442}$.