Real Sequence with Nonzero Limit is Eventually Nonzero

Theorem

Let $\sequence {x_n}$ be a real sequence.

Let $\sequence {x_n}$ converge to $a \ne 0$.

Then:

$\exists N \in \N: \forall n \ge N: x_n \ne 0$

That is, eventually every term of $\sequence {x_n}$ becomes non-zero.

Proof

Suppose $a > 0$.

By Sequence Converges to Within Half Limit:

$\exists N \in \N: \forall n > N: x_n > \dfrac a 2 > 0$

Now suppose $a < 0$.

By Sequence Converges to Within Half Limit:

$\exists N \in \N: \forall n > N: x_n < \dfrac a 2 < 0$

This shows that if $a \ne 0$:

$\exists N \in \N: \forall n > N: x_n \ne 0$

$\blacksquare$

Also see

• Limit of Positive Real Sequence is Positive