Real exponential function/Continuity and image/Fact/Proof

Proof

The continuity follows from fact, since the exponential function is defined with the help of a power series. Due to fact  (4), the image lies in ${}\mathbb {R} _{+}$, and the image is, because of the intermediate value theorem, an interval. The unboundedness of the image follows from fact  (3). This implies, because of fact  (2), that also arbitrary small positive real numbers are obtained. Thus the image is ${}\mathbb {R} _{+}$. Injectivity follows from fact  (6), in connection with exercise.

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