It is rather straight forward to show that $L_p$ is complete for $p\geqslant 1$, but I am having trouble showing the same thing when $p<1$. For the former case I have shown that every absolutely convergent sequence converges by constructing a a function in $L_p$ but bigger than the series and used the dominated convergence theorem (I can also do a similar thing using Cauchy sequences rather than absolutely convergent series). The problem is that in showing that my upper bound function is an element of $L_p$ I have used the triangle inequality which I cannot do for $p<1$. Does anyone have any ideas of a way around this?
I noticed there are similar questions to this already, but they have either not been answered or closed.