# Proof of inequality using Cauchy–Schwarz inequality

How I can prove this inequality? How can I prove it using Cauchy–Schwarz inequality?

Let $$a_1,...,a_n,b_1,....,b_n$$ are any real numbers.

$$\sqrt {\sum_{k=1}^n (a_k+b_k)^2)} \leq \sqrt {\sum_{k=1}^n {a_k}^2} \sqrt {\sum_{k=1}^n {b_k}^2}$$

• Do you mean $\sqrt{\sum_{k=1}^n a_k+b_k}=\sqrt{\sum_{k=1}^n a_k^2}+\sqrt{\sum_{k=1}^n a_k^2}$? Mar 20, 2019 at 18:19

This is false when $$a_1 = b_1= 1$$ and $$a_k = b_k = 0$$ for $$k \ne 1$$.