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 $n \in \N$:
\(\ds \size {x_n - L}\) | \(=\) | \(\ds x_n - L\) | $x_n \ge 0 > L$ | |||||||||||
\(\ds \) | \(\ge\) | \(\ds -L\) | $> 0$ |
This contradicts Definition of Convergent Real Sequence.
Hence we must have $L \ge 0$.
$\blacksquare$