If ${{a}}_1,{{a}}_2,\ldots,{{a}}_n$ are any real numbers and $n$ is any positive integer, then

$$ \begin{aligned} &\textbf{To prove: } \quad n\sum_{i=1}^{n} a_i^2 \;\ge\; \left(\sum_{i=1}^{n} a_i\right)^2 \\[6pt] &\text{(RMS}\ge\text{AM)}:\quad \sqrt{\frac{\sum_{i=1}^{n} a_i^2}{\,n\,}} \;\ge\; \frac{\sum_{i=1}^{n} a_i}{\,n\,} \\[6pt] &\Rightarrow\; \frac{\sum_{i=1}^{n} a_i^2}{\,n\,} \;\ge\; \left(\frac{\sum_{i=1}^{n} a_i}{\,n\,}\right)^2 \;\Rightarrow\; n\sum_{i=1}^{n} a_i^2 \;\ge\; \left(\sum_{i=1}^{n} a_i\right)^2. \\[6pt] &\text{Equality iff } a_1=a_2=\cdots=a_n. \end{aligned} $$