I want to prove that $\sin(x)$ is continous in $\mathbb R$, the definition that I have of $\sin(x)$ is the definition of the Spivak's book (the geometric one), but maybe using some other definition would make the prove easier?
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Which definition is used in Spivak? The geometric one? If you have that and the addition formula, then you know $|\sin(x+h)-\sin(x)| \leq |\sin(x)(\cos(h)-1)|+|\cos(x)\sin(h)| \leq |\sin(x)(\cos(h)-1)|+|h|$. So the only difficulty is controlling the first term. In fact you can be even blunter: $|\sin(x+h)-\sin(x)| \leq |\cos(h)-1|+|h|$. So the problem just reduces to showing $\cos$ is continuous at $0$. – Ian Nov 25 '15 at 22:45
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2Please state the definition, since not everyone in the world uses the same text books. – Graham Kemp Nov 25 '15 at 22:46
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Yes, the geometric one. – Giotaker Nov 25 '15 at 22:47