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  "body": "\n## Statement\n\nInduction is often used to prove inequalities. As an example, we prove that $\\left|\\sin nx\\right| \\leq n\\left|\\sin x\\right|$ for any [[real number|08J2M1JYAG]] $x$ and natural number $n$.\n\nAt first glance, it may appear that a more general version, $\\left|\\sin nx\\right| \\leq n\\left|\\sin x\\right|$ for *any* real numbers $n,x$, could be proven without induction; but the case $n = \\frac{1}{2},\\, x=\\pi$ shows it may be false for non-integer values of $n$. This suggests we examine the statement specifically for *natural* values of $n$, and induction is the readiest tool.\n",
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