Question Number 181699 by Shrinava last updated on 28/Nov/22
![Prove it by mathematical induction: ∣ Σ_(j=1) ^n x_j ∣ ≤ Σ_(j=1) ^n sin x_j ; x_j ∈ [ 0 , π ]](Q181699.png)
$$\mathrm{Prove}\:\mathrm{it}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}: \\ $$$$\mid\:\:\underset{\boldsymbol{\mathrm{j}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{x}_{\boldsymbol{\mathrm{j}}} \:\:\mid\:\:\leqslant\:\:\underset{\boldsymbol{\mathrm{j}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{sin}\:\mathrm{x}_{\boldsymbol{\mathrm{j}}} \:\:\:\:\:;\:\:\:\:\:\mathrm{x}_{\boldsymbol{\mathrm{j}}} \:\in\:\left[\:\mathrm{0}\:,\:\pi\:\right] \\ $$
Commented by mr W last updated on 28/Nov/22
$${what}\:{if}\:{x}_{{j}} =\frac{\pi}{\mathrm{2}}? \\ $$$${LHS}={n}×\frac{\pi}{\mathrm{2}} \\ $$$${RHS}={n}×\mathrm{1}={n} \\ $$$${LHS}>{RHS}\:!\: \\ $$$$\Rightarrow{question}\:{is}\:{wrong}! \\ $$