we define $ l_p=\{x=(x_n) : \sum_{i=1}^\infty |x_n|^p < \infty\} (p\in [1,\infty) ) $
I tried to prove that $$\left\|x\right\|=\left(\sum_{i=1}^\infty\left|x_i\right|^p\right)$$ is a p norm. I tried 1. and 2. conditon but I can't prove triangle inequality for $p$-norm. I tried to use minkowski inquality but I can't use it because there is no $(1/p)$ on ther series. I also tried to prove it like this: $$\left(\sum_{i=1}^\infty\left|x_i+y_i\right|^p\right)\leq 2^p \left(\sum_{i=1}^\infty\left|x_i\right|^p+|y_i|^p\right)\leq 2^p\left(\sum_{i=1}^\infty\left|x_i\right|^p\right)+2^p\left(\sum_{i=1}^\infty\left|y_i\right|^p\right)\neq \left\|x\right\|+ \left\|y\right\|$$ but this is not working. How to prove triangle inequality for this p-norm?