# Real Sequence with Nonzero Limit is Eventually Nonzero

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## 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