# Math Help - Simple question regarding sequences and subsequences

1. ## Simple question regarding sequences and subsequences

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

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

Also, does every sequence necessarily have a subsequence?

2. Originally Posted by Ares_D1
I'm sure I'm overlooking or misunderstanding some obvious detail here, but nevertheless:

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

Also, does every sequence necessarily have a subsequence?
Dear Ares_D1,

If you consider the definition of the limit of a sequence, it is defined for infinite sequences only.( Limit of a sequence - Wikipedia, the free encyclopedia) Therefore for a finite sequence there is no such thing as a limit.