Proof

By hypothesis, let $p\in[1,\infty)$ and $x=(x_1,x_2,\ldots x_n)$ and $y=(y_1,y_2,\ldots y_n)$ be two vectors of a vector space $V$ over the field of real numbers $\mathbb R$ or the field of complex numbers $\mathbb C$.
If $p=1$, the inequality follows from the triangle inequality.
Let $p > 1$ and define $q$ by $\frac 1p+\frac 1q=1,$ i.e. $q=\frac{p}{p-1}.$
Consider the vector $z\in\mathbb C^n$ (or, in real case, $z\in\mathbb R^n$) with $z_\nu:=|x_\nu+y_\nu|^{p-1}$ for $\nu=1,\ldots,n.$
Then we get $z_\nu^q=|x_\nu+y_\nu|^{q(p-1)}=|x_\nu+y_\nu|^{p}$ for $\nu=1,\ldots,n,$ and this yields for the q-norm of the vector $z$ $$\begin{array}{rcl}||z||_q&=&\left(\sum_{\nu=1}^n|z_\nu|^q\right)^{1/q}\\ &=&\left(\sum_{\nu=1}^n|x_\nu+y_\nu|^{p}\right)^{1/q}\\ &=&\left(\sum_{\nu=1}^n|x_\nu+y_\nu|^{p}\right)^{1/p\cdot p/q}\\ &=&||x+y||_p^{p/q}.\end{array}$$
We can now estimate:
- $\sum_{\nu=1}^n|x_\nu+y_\nu||z_\nu|\le \sum_{\nu=1}^n|x_\nu z_\nu|+\sum_{\nu=1}^n|y_\nu z_\nu|$ by the triangle inequality, and
- $\le (||x||_p+||y||_p)||z||_q$ by the Hölder's inequality.
This yields by the definition of the vector $z$ $$\begin{array}{rcl}||x+y||_p^p&=&\sum_{\nu=1}^n|x_\nu+y_\nu|^p\\ &=&\sum_{\nu=1}^n|x_\nu+y_\nu||x_\nu+y_\nu|^{p-1}\\ &=&\sum_{\nu=1}^n|x_\nu+y_\nu||z_\nu|\\ &\le&(||x||_p+||y||_p)||z||_q\\ &=&(||x||_p+||y||_p)||x+y||_p^{p/q}.\end{array}$$
Since $p-\frac pq=1$, we get (dividing both sides of the last inequality by $||x+y||_p^{p/q}$) $$||x+y||_p\le ||x||_p+||y||_p.$$
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Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983