(related to Proposition: Continuity of Cosine and Sine)

Let \((x_n)_{n\in\mathbb N}\) be a convergent real sequence with \[\lim_{n\to\infty}x_n=a\] for some real number \(a\in\mathbb R\). We want to show that cosine \(\cos:\mathbb R\mapsto \mathbb R\) and sine \(\sin:\mathbb R\mapsto \mathbb R\) are continuous real functions, which means by definition that \[\begin{array}{rcl}\lim_{n\to\infty}\cos(x_n)&=&\cos(a)\\\lim_{n\to\infty}\sin(x_n)&=&\sin(a).\end{array}\]

We use the fact that real numbers are embedded in the complex numbers. Therefore, we can consider all real sequence members \(x_n\) to be special complex numbers with \(x_n=\Re( x_n)\). Then \((x_n)_{n\in\mathbb N}\) is a convergent complex sequence. Due to the rule of multiplying a complex sequence by a complex number (here, we multiply all \(x_n\) by the imaginary unit \(i\), we get the result


Because the complex exponential function is continuous, we get in particular on the unit circle the next result


We also know that a complex sequence is convergent if and only if its real and imaginary parts are convergent. It follows from the Euler's formula that


By comparing the real and imaginary parts of the last equation, we get the desired result:


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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983