In Serre's Local fields, Chapter One, Section One. I have difficulty proving that $v(x+y)\geq $inf$(v(x),v(y))$. Here is my attempt. Take $x=\pi^{n_{1}}u$ and $y=\pi^{n_{2}}u'$, then assume that $n_{2}>n_{1}$, we have $\pi^{n_{1}}(u+\pi^{n_{2}-n_{1}}u')$. Consequently, $v(\pi^{n_{1}}u+\pi^{n_{2}}u')=\pi^{n_{1}}(u+\pi^{n_{2}-n_{1}}u')=v(\pi^{n_{1}})+v(u+\pi^{n_{2}-n_{1}}u')$. However, I fail to do any successful calculation about $v(u+\pi^{n_{2}-n_{1}}u')$. I think that I'm on the wrong path.
