Infinity norm is actually a norm : triangle inequality

I have to prove the following assertion : Let $$V$$ be a finit dimentional vector space with dimension $$n$$ over the field $$K$$ which is the field of real numbers or complex numbers. Let the map defined by: $$\mid \mid . \mid \mid _{\infty} \quad : v \mapsto \mid \mid v \mid \mid_{\infty}=\max\{\mid v_1\mid,...,\mid v_n \mid \} \quad \forall v \in V$$ where $$v_k \quad k=1,...,n$$ are the componant of the vector $$v$$. Show that this map is a norm.

• I proved the posivity of this norm.
• I also proved that : $$\mid \mid \lambda v \mid \mid _{\infty}=\mid \lambda \mid \mid \mid v \mid \mid \quad \forall v \in V, \forall \lambda \in K$$

Now, I have to prove the triangle inequality. I have already done something but i am not sure what i have done is correct : I have to prove that : $$\forall v, w \in V \quad \mid \mid v+w \mid \mid _{\infty}\leq \mid \mid v \mid \mid_{\infty} + \mid \mid w \mid \mid _{\infty}$$ I know that in the real or complex numbers, we have this inequality for all $$x$$ and $$y$$ : $$\mid x+y\mid \leq \mid x \mid +\mid y\mid$$ Therefore : $$v=(v_1,...,v_n)$$, $$w=(w_1,...,w_n)$$ : $$\forall k \in {1,...,n} \quad \mid v_k + w_k \mid \leq \mid v_k \mid +\mid w_k \mid$$ Then, $$\max\{\mid v_1 +w_1\mid,... \mid v_n +w_n\mid \} \leq \max\{\mid v_1 \mid +\mid w_1 \mid ,...,\mid v_n\mid+\mid w_n \mid \}$$ Now the problem is : can i say that $$\max\{\mid v_1 \mid +\mid w_1 \mid ,...,\mid v_n\mid+\mid w_n \mid \} \leq \max\{\mid v_1\mid ,...\mid v_n\mid \}+\max\{\mid w_1\mid ,...\mid w_n\mid \}$$ which would prove the inequality ? Is it trivial ? If it is not, i don't know how to prove the last inequality...

It is very easy. All you need is $$|v_i|+|w_i| \leq \max \{|v_1|,...,|v_n|\}+\max \{|w_1|,...,|w_n|\}$$ for each $$i$$. This is true, right?
It's trivial: for each index $$j$$ we have $$|x_j+y_j|\le |x_j|+|y_j|\le\|x\|_\infty+\|y\|_\infty$$