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

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