Coordinate projections in $l^p$ are Lipschitz continuous

Question

Prove that for all $p\in[1,\infty]$ the k-th projection $\pi_k:l_p\to \mathbb R$ $\pi_k(x)=x_k$ is Lipschitz continuous

I need to prove that for all $x,y\in l_p$ $\exists K>0$ such that $|x_k-y_k|\le K||x-y||_p$

I´m having problems in how can bound $|x_k-y_k|$ in terms of the p norm

I would really appreciate if can help me with this problem, any ideas would be highly appreciated

Answer 1

It's actually quite easy. note that $|x_k - y_k| = \big( |x_k - y_k|^p \big)^{\frac 1p} \le \big( |x_k - y_k|^p + \cdots \big)^{\frac 1p} $ -- John Ma