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## Homework Statement

What I want to show is this:

∫|x+y| ≤ ∫|x| + ∫|y|

## Homework Equations

|x+y| ≤ |x| + |y|

## The Attempt at a Solution

So I thought if I used the triangle inequality I could get to something along the lines of:

Lets g belong to the real numbers

∫|x+y| = ∫|x+g-g+y|≤ ∫|x+g| + |y-g|= ∫|x+g| + ∫|y-g|

As g belongs to the reals it can be zero meaning ∫|x+y| ≤ ∫|x| + ∫|y|.

Now the problem with this is that is uses the triangle inequality and I have no idea if the triangle inequality works this way, and if it does I need to prove it, and I have no idea about where to start that from.