For consecutive primes $a\lt b\lt c$, prove that $a+b\ge c$.


I cannot find a counter-example to this. Do we know if this inequality is true? Alternatively, is this some documented problem (solved or unsolved)? 



For consecutive primes $a\lt b\lt c$, prove that $a+b\ge c$.
Yes, this is true. In 1952, [Nagura](https://projecteuclid.org/euclid.pja/1195570997) proved that for $n \geq 25$, there is always a prime between $n$ and $\frac{6}{5} n$. Thus, let $p\_k$ be a prime at least $25$. Then $p\_k+p\_{k+1} > 2p\_k$. But by Nagura's result we have that $p\_{k+2} \leq \frac{36}{25} p\_k < 2p\_k$. Finally, one can easily check by hand that the result holds for small primes.


