Exponential of Real Number is Strictly Positive/Proof 2
Jump to navigation
Jump to search
Theorem
Let $x$ be a real number.
Let $\exp$ denote the (real) exponential function.
Then:
- $\forall x \in \R : \exp x > 0$
Proof
This proof assumes the limit definition of $\exp$.
That is, let:
- $\ds \exp x = \lim_{n \mathop \to \infty} \map {f_n} x$
where $\map {f_n} x = \paren {1 + \dfrac x n}^n$
First, fix $x \in \R$.
Let $N = \ceiling {\size x}$, where $\ceiling {\, \cdot \,}$ denotes the ceiling function.
Then:
\(\ds \exp x\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \map {f_n} x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \map {f_{n + N} } x\) | Tail of Convergent Sequence | |||||||||||
\(\ds \) | \(\ge\) | \(\ds \map {f_{n + N} } x\) | Exponential Sequence is Eventually Increasing and Limit of Bounded Convergent Sequence is Bounded | |||||||||||
\(\ds \) | \(>\) | \(\ds 0\) | Corollary to Exponential Sequence is Eventually Increasing |
This article needs to be linked to other articles. In particular: Corollary to Exponential Sequence is Eventually Increasing does not actually exist. The page it gets sent to does not give that result. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code. |
$\blacksquare$