Let $l_p=\{(x_n)\in\mathbb{R}^\mathbb{N}: \sum_{n=0}^\infty |x_n|^p<\infty\}$ and consider the following norm in $l_p$: $$\|x\|_p=\left(\sum_{n=0}^\infty|x_n|^p\right)^{1/p}$$ for $x=(x_n)_{n\in\mathbb{N}}$. I'm studying this norm, and I have to prove that the inequality $\|x+y\|_p<\|x\|_p+\|y\|_p$ holds for $x\neq \lambda y$, $\lambda>0$.
My attempt: the restriction is needed because if $y=\lambda x$ then the equality $$\|x+y\|_p=\|(1+\lambda)x\|_p=|\|x\|_p+\lambda\|x\|_p=\|x\|_p+\|y\|_p$$ holds ever.
Now I do the particular case $y=-\lambda x$, for $0<\lambda\leq 1$ (and $x\neq 0$) then: $$\|x+y\|_p=\|x-\lambda x\|_p= \|(1-\lambda)x\|_p=(1-\lambda)\|x\|_p=\|x\|_p-\lambda\|x\|_p<\|x\|_p+\|y\|_p$$ (the ''$<$'' is because $\|x\|_p>0 $)
Now if $\lambda>1$, I found that the inequality holds again with '''<'' just following a similar approach.
But how can I proceed in the general case? Thanks in advance!