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  "body": "\n## Example\n\nLet $r$ be a real number with $|r| \u003c 1$.\nThen $r^n \\to 0$ as $n \\to \\infty$.\nFor $r = 0$ this is clear. For $0 \u003c |r| \u003c 1$, write $1/|r| = 1 + h$ with $h \u003e 0$.\nBy Bernoulli's inequality, $(1 + h)^n \\ge 1 + n h \u003e n h$ for each $n \\ge 1$.\nThus $0 \\le |r^n| \u003c 1/(n h)$, and the [[squeeze theorem|0XMF00T4XQ]] gives $|r^n| \\to 0$, so $r^n \\to 0$.\n\n## Remarks\n\nThe [[geometric sequence|0MNM54WB99]] $a r^n$ then tends to $0$ for each real $a$.\nFor $r = 1$ the sequence is constant. For $r = -1$ it is $1, -1, 1, \\dots$, which diverges.\nFor $|r| \u003e 1$ the sequence $|r^n|$ [[tends to infinity|0TZK91NQE3]].\n\n## Explanations\n\n### L1 human\n\nMultiply a number by $1/2$ again and again: $1, 1/2, 1/4, 1/8, \\dots$.\nThe result gets closer and closer to $0$. The same happens for each factor between $-1$ and $1$.\n",
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