# Proof

• By hypothesis, $[a,b]$ is a closed real interval, $p\ge 1$ is a real numbers, and $f,g:[a,b]\to\mathbb R$ are two Riemann-integrable functions.
• Since $f,g$ are Riemann-integrable, so are the functions $f+g,$ $f^p,$ and $g^p.$
• Let $n\ge 1$ and $h:=\frac{b-a}n.$
• The Riemann sum converges to the Riemann integral, for suitably big $n$ and the mash points $\xi_k:=h*k+a,$ $k=1,\ldots,n.$
• According to the Minkowski's inequality we get with the p-norms of the vectors $\phi:=(f(\xi_1),f(\xi_2),\ldots,f(\xi_n))$, $\psi:=(g(\xi_1),g(\xi_2),\ldots,g(\xi_n))$: $$\begin{eqnarray}\left(\sum_{k=1}^n|f(\xi_k)+g(\xi_k)|^p\right)^{\frac 1p}\cdot h&\le &\left(\sum_{k=1}^n|f(\xi_k)|^p\right)^{\frac 1p} h+\left(\sum_{k=1}^n|g(\xi_k)|^p\right)^{\frac 1p} h\nonumber\\&=&||\phi||_p\cdot h+||\psi||_p\cdot h.\nonumber\end{eqnarray}$$
• For $n\to\infty$ we get with the integral p-norms the inequality $$||f+g||_p\le ||f||_p+||g||_p.$$

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983