Proof of an inequality with the triangle inequality

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How can I prove this inequality using the triangle inequality?

$|a-b| \leq |a-c| + |c-b|$

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$$|a-b|=|(a-c)+(c-b)|\leq |a-c|+|c-b|$$

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$$\vert (a-c)+(c-b) \vert = \vert a-b \vert \leq \vert a-c \vert + \vert c-b \vert$$