Zero is not a Limit Point of Sequence of Reciprocals and Reciprocals + 1

Theorem
Let $\struct {\R, \tau}$ denote the real number line under the usual (Euclidean) topology.

Let $\sequence {a_n}$ denote the sequence in $\struct {\R, \tau}$ defined as:

Then $0$ is not a limit of $\sequence {a_n}$.

Proof
The open interval $\openint {-1} 1$ contains $0$, and also contains all terms of $\sequence {a_n}$ with odd indices greater than $1$.

However, all terms of $\sequence {a_n}$ with even indices are outside $\openint {-1} 1$.

Hence $0$ cannot be a limit of $\sequence {a_n}$.

Also see

 * Accumulation Point of Sequence of Reciprocals and Reciprocals + 1 where it is shown that $0$ is an accumulation point of $\sequence {a_n}$.