This page is about subsequences of a sequence. A subsequence
of a sequence (_{
)
} is an infinite collection of numbers
from (_{
)
} in the same order that they appear in that
sequence.

The main theorem on subsequences is that
every subsequence of a convergent sequence (_{
)
}
converges to the same limit as (_{
)
}.
This (together with the Theorem on Uniqueness of Limits)
is the main tool in showing a sequence *does not converge*.

It is harder to use subsequences to prove a sequence *does* converge
and much easier to make mistakes in this direction.
Just knowing a sequence (_{
)
} has a
convergent subsequence says nothing about the convergence of
(_{
)
}. We discuss these topics with an example in
another web page.

If (_{
)
} is a sequence, such as ** _{1=1=1}**,

subsequenceis rather simple and very natural. Unfortunately there are some difficulties with notation that tends to obscure the main idea and can make the manipulation of subsequences a little more tricky.

Definition of subsequence.

Suppose (_{
)
} is a sequence and
_{
)
} and

with elements

The sequence (** _{
)
}** is called a
subsequence of (

Example.

The ideas of selecting infinitely many terms from a sequence

and using an increasing function to enumerate a selection of terms
are equivalent. If we have an infinite set of terms we can enumerate
them with a function

Lemma.

Let

**Proof.**

By induction on

Theorem on Subsequences.

**Proof.**

Part of the power of this theorem is that the subsequence (** _{
)
}**
converges to

Example.

We use these ideas to re-prove the assertion that the sequence _{
=(-1)
}
doesn't converge. Define subsequences ** _{
=
2+1
}** and

Example.

Consider the sequence defined by
_{
=()
}. This is proved to be non-convergent as follows.

First a subsequence (** _{
)
}** is selected consisting of all

Next a subsequence (_{
)
}
_{
}
for which

Finally, suppose _{
}. Then by the Theorem on Subsequences
both ** _{
}** and

Be careful to note the _{
0
}