Limit of Positive Real Sequence is Positive

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Theorem

Let $\sequence {x_n}$ be a sequence of positive real numbers.

Let $x_n$ converge to $L$.


Then $L \ge 0$.


Proof

Aiming for a contradiction, suppose $L < 0$.

Then for any $n \in \N$:

\(\displaystyle \size {x_n - L}\) \(=\) \(\displaystyle x_n - L\) $x_n \ge 0 > L$
\(\displaystyle \) \(\ge\) \(\displaystyle -L\) $> 0$

This contradicts Definition of Convergent Real Sequence.


Hence we must have $L \ge 0$.

$\blacksquare$