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 $L \ge 0$.

$\blacksquare$