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**Ares_D1** I'm sure I'm overlooking or misunderstanding some obvious detail here, but nevertheless:

It can be shown that for any sequence $\displaystyle (a_n)$ that converges to some limit L, any *subsequence* $\displaystyle (a_{nj})$ of $\displaystyle (a_n)$ also converges to L. Correct? But I don't quite understand this. Suppose we have the (arbitrary) sequence $\displaystyle a_n = \{1, 2, 3, 4, 5 \}$, which converges to 5. But then $\displaystyle a_{nj} = \{1, 3 \}$ is a subsequence of $\displaystyle (a_n)$, which converges to 3, no? What obvious detail am I missing here?

Also, does every sequence necessarily have a subsequence?