Inequality similar to Minkowski

Question

Prove that $(|x_1-z_1|^p+|x_2-z_2|^p)^{\frac{1}{p}} \le (|x_1-y_1|^p + |x_2-y_2|^p)^{\frac{1}{p}}+(|y_1-z_1|^p + |y_2-z_2|^p)^{\frac{1}{p}}$ The form is very close to Minkowski's inequality,but I can't really show this.

Answer 1

First of all, you need $p\geq1$.

If so, it's just Minkowski because by the triangle inequality $$|x_1-y_1|+|y_1-z_1|\geq|x_1-z_1|$$ and

$$|x_2-y_2|+|y_2-z_2|\geq|x_2-z_2|.$$