Question about the $L^p$ being complete.

The inequality $(1+t)^p\leq 1+t^p$, which can be obtained by showing that $g(t)=(1+t)^p-t^p$ ($t>0$) is monotone decreasing implies that$$(a+b)^p\leq a^p+b^p,\quad 0<p<1,\,a,b\geq0.$$It then follows that $d_p(f,g) = \int |f-g|^p$ defines a metric on $L_p$.I was wondering if with $(L^p,d_p)$ is complete.I think the answer would be yes, but I cannot come up with a proof. It seems to me that it would be in the same spirit of proving the case for $p \geq 1$ but I am unsure.