Prove inequality for vectors in $\mathbb{R}^{n}$
Solution 1:
$$\left\| \sum_{j=1}^n y_j\right\|^2 \le \left(\sum_{j=1}^n\|y_j\|\right)^2 \le \left(\sum_{j=1}^n 1\right)\left(\sum_{j=1}^n \|y_j\|^2\right) = n\sum_{j=1}^n \|y_j\|^2\le \frac{n(n+1)}{2}\sum_{j=1}^n \|y_j\|^2$$