Limit of Positive Real Sequence is Positive

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 $L \ge 0$.

$\blacksquare$