The theorem of sign permanence for sequences states that if a real sequence \( a_n \) converges to a limit \( L \neq 0 \), there exists an index \( N \) beyond which all terms of the sequence have the same sign as \( L \). In other words:

\[ \lim_{n\to\infty} a_n = L > 0 \, \implies \, \exists N \in \mathbb{N} \, : \, \forall n \geq N \, , \, a_n > 0\]

On the other hand, if \( L < 0\), then:

\[ \lim_{n\to\infty} a_n = L < 0 \, \implies \, \exists N \in \mathbb{N} \, : \, \forall n \geq N \, , \, a_n < 0 \]

By definition,

\[ \lim_{n\to\infty} a_n = L \, \iff \, \exists N \in \mathbb{N} \, : \, \forall n \geq N \, , \, |a_n - L| < \epsilon \]

In particular, by choosing \( \epsilon = \frac{L}{2} \), we have

\[ L - \frac{L}{2} < a_n < L + \frac{L}{2} \]

Now, observe that:

• If \( L > 0 \), then 

\[ \left ( L - \frac{L}{2} \right ) = \frac{L}{2} < a_n < \frac{3L}{2} = \left ( L + \frac{L}{2} \right ) \qquad \forall n \geq N\]

• If \(L = - |L| < 0\), then 

\[ \left ( -|L| - \frac{|L|}{2}\right ) = -\frac{3|L|}{2} < a_n < -\frac{|L|}{2} = \left ( -|L| + \frac{|L|}{2} \right ) \qquad \forall n \geq N \]

In both cases, starting from \( N \), the terms of the sequence \( a_n \) have the same sign as \( L \).