# an inequality like the triangle inequality

Its easy question, but I cannot find the name of the inequality. Please provide me with it.

I am doing the following, let $a$ be $n$-dimentional vector. Let $b_i, i=1, \ldots, n$ be positive numbers. Then, $$\left(\left|\sum_{i=1}^n a_i\right|^p\right)^{1/p}=\left(\left|\sum_{i=1}^n(a_i-b_i)+\sum_{i=1}^nb_i\right|^p\right)^{1/p}\leq \left(\left|\sum_{i=1}^n(a_i-b_i)\right|^p\right)^{1/p}+\left(\left|\sum_{i=1}^nb_i\right|^p\right)^{1/p}$$ Which inequality did I use here?

Thank you.

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The way you wrote it, this is just the triangle inequality in $\mathbb{R}$. Maybe you misplaced some brackets? – IHaveAStupidQuestion Nov 4 '12 at 16:37
Indeed, notice that the exponents just cancel out, i.e. $(x^p)^{1/p}=x$. – nonpop Nov 4 '12 at 17:15