How can I prove the inequality for integrals of integrable vectors?

[dataphile]

Homework Statement

Let $f: X \rightarrow \mathbb{R}^{n}.$ We say that $\textbf{f} = (f_{1}, ..., f_{n})$ is integrable if each component $f_{k}$ is integrable, and define $\int \textbf{f} d\mu = (\int f_{1} d\mu, ..., \int f_{n} d\mu).$ Show that if $\textbf{f}$ is integrable, then $| \int \textbf{f} d\mu | \leq \int | \textbf{f} | d\mu.$

Homework Equations

The chapter discusses various inequalities for integrals, including Jensen's inequality, Holder's inequality, the definition of an $L^{p}$ norm, Minkowski's inequality, etc.

The Attempt at a Solution

The inequality is saying that the magnitude of the integral of a vector is less than or equal to taking the integral of the magnitude of the vector. To do the formal proof, I've experimented with using the triangle inequality, but can't seem to get everything in the right place. The closest I have is this:

$$| \int \textbf{f} d\mu | \leq \sum_{i=1}^{n} | \int f_{i} d\mu |$$ (by the triangle inequality)

$$\sum_{i=1}^{n} | \int f_{i} d\mu | \leq \sum_{i=1}^{n} \int |f_{i}| d\mu = \int \sum_{i=1}^{n} |f_{i}| d\mu$$

So I end up with something in the $L^1$ norm and can't make the last steps to get to $$\int | \textbf{f} | d\mu = \int ((\sum_{i=1}^{n} f_{i}^2)^\frac{1}{2}) d\mu$$

I would be grateful if someone could just point me in the right direction.

 

[sunjin09]

All your inequalities are valid for all Lp norms, 1,2, ... \inf

 

[dataphile]

I know that the inequality will hold true for any $L^p$ norm, but I am still having trouble finishing the proof for the particular case of the $L^2$ norm.