I'm trying to prove that for $a,b\in \mathbb{R}^+$, if $a<b$, then $a^{1/n} < b^{1/n}$. I'm naturally proceeding by induction on $n$, with it being trivial for $1$. I've proven it separately before with $n=2$ as well, and I'm just gong to assume it true here for convenience. My problem is in the induction step, having it assumed to be true for some fixed $n$. I need to show $a^{1/(n+1)} < b^{1/(n+1)} $. I'm having a complete brain fart. Can anyone help?