Properties of Real Sine Function

Theorem
Let $x \in \R$ be a real number.

Let $\sin x$ be the sine of $x$.

Then:
 * $(1): \quad \sin x$ is continuous on $\R$.
 * $(2): \quad \sin x$ is absolutely convergent for all $x \in \R$.
 * $(3): \quad \sin 0 = 0$.
 * $(4): \quad \sin \left({-x}\right) = -\sin x$.

Proof
Recall the definition:


 * $\displaystyle \sin x = \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$


 * $(1): \quad$ Continuity of $\sin x$:


 * $(2): \quad$ Absolute convergence of $\sin x$:

For:
 * $\displaystyle \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!}$

to be absolutely convergent we want:
 * $\displaystyle \sum_{n=0}^\infty \left|{\left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!}}\right| = \sum_{n=0}^\infty \frac {\left|{x}\right|^{2n+1}}{\left({2n+1}\right)!}$

to be convergent.

But:
 * $\displaystyle \sum_{n=0}^\infty \frac {\left|{x}\right|^{2n+1}}{\left({2n+1}\right)!}$

is just the terms of:
 * $\displaystyle \sum_{n=0}^\infty \frac {\left|{x}\right|^n}{n!}$

for odd $n$.

Thus:
 * $\displaystyle \sum_{n=0}^\infty \frac {\left|{x}\right|^{2n+1}}{\left({2n+1}\right)!} < \sum_{n=0}^\infty \frac {\left|{x}\right|^n}{n!}$

But:
 * $\displaystyle \sum_{n=0}^\infty \frac {\left|{x}\right|^n}{n!} = \exp \left|{x}\right|$

from the Taylor Series Expansion for Exponential Function of $\left|{x}\right|$, which converges for all $x \in \R$.

The result follows from the Squeeze Theorem.


 * $(3): \quad \sin 0 = 0$:

Follows directly from the definition:
 * $\displaystyle \sin 0 = 0 - \frac {0^3} {3!} + \frac {0^5} {5!} - \cdots = 0$


 * $(4): \quad \sin \left({-x}\right) = -\sin x$:

From Sign of Odd Power, we have that:
 * $\forall n \in \N: -\left({x^{2n+1}}\right) = \left({-x}\right)^{2n+1}$

The result follows from the definition.