Prove ln(x) is uniformly continuous on (1,2)

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Proving the function ln(x) is uniformly continuous on (1,2) by first arguing it's uniformly continuous on [1,2].
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ln(x) is continuous on [1;2] -> uniformly continuous on [1;2]

МаксимАндреев-щб
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how to prove that it's uniform continuous on (0, +infty) ?

Teob