Real Analysis – General Proof for the Triangle Inequality

Question

I am trying to prove: $P(n): |x_1| + \cdots + |x_n| \leq |x_1 + \cdots +x_n|$ for all natural numbers $n$. The $x_i$ are real numbers.

Base: Let $n =1$:
we have $|x_1| \leq |x_1|$ which is clearly true

Step: Let $k$ exist in the integers such that $k \geq 1$ and assume $P(k)$ is true.

This is where I am lost. I do not see how to leverage the induction hypothesis.

Here is my latest approach:

Can you do the following in the induction step: Let $Y$ = |$x_1$ +…+$x_n$| and Let $Z$ = |$x_1$| +…+ |$x_n$| Then we have: |$Y$ + $x_n+1$| $\leq$ $Z$ + |$x_n+1$|. $Y$ $\leq$ $Z$ from the induction step, and then from the base case this is just another triangle inequality. End of proof.

Answer 1

As @ivan indicates, the inequality is reversed - it should be

$$ |x_1 + x_2 + \dots + x_n| \leq |x_1| + |x_2| + \dots + |x_n| $$

As the base case for induction, you need to show (or assert? can you take the "basic" triangle inequality for granted?)

$$ |x_1 + x_2| \leq |x_1| + |x_2|. $$

Hint:

One way to do this is to show $(|a + b|)^2 \leq (|a| + |b|)^2$ by expanding the LHS and using $ab \leq |a||b|$.

Then, for induction, assume

$$ |x_1 + x_2 + \dots + x_n| \leq |x_1| + |x_2| + \dots + |x_n| $$

and show

$$ |x_1 + x_2 + \dots + x_n + x_{n+1}| \leq |x_1| + |x_2| + \dots + |x_n| + |x_{n+1}|$$

using the induction hypothesis and the base case.