We have seen how to write down statements using quantifiers and some of the rules for manipulating quantifier in proofs. We have also see the definition of convergence of a sequence, and noted that it was rather complicated. I said that these complications were in fact necessary, and nothing simpler will do. I'm going to partially justify this claim now by looking at a modification of the definition of convergence that at first sight looks reasonable, but which turns out to be very wrong.

Recall that a sequence (_{
)
} converges to 0 if

We are going to see the effect of changing round the first
two quantifiers. Say that a sequence (_{
)
}
verconges to 0 if

At first sight, this looks a reasonable idea, and
might also capture the right notion of convergence

.
So our question is, what does it mean to say a sequence
verconges to 0, and is it the same thing as
converging to 0? To answer this, we first look at
some examples.

Recall the integer-part function,
_{
=100/
}.

Proposition.

(_{
)
} verconges to zero.

**Proof.**

We must prove that _{
}<

Remark.

This statement starts with a

**Subproof.**

Hence _{
}<

That's good. At least there is *some sequence* that we know
about now that converges to 0 and also verconges to 0.
But sadly, there are examples of sequences that converge to 0
but don't verconge to 0.

Now consider the sequence (** _{
)
}** with terms

Proposition.

(** _{
)
}** does not verconge to zero.

**Proof.**

Remark.

The first step is to push the not

inside the quantifiers
in not (

. When we do
this carefully we realise that we must prove that ** _{
)
}** verconges to 0

As the first quantifier is a _{
}

**Subproof.**

Let

**Subproof.**

Let

So (_{
}

So _{
}

So _{
}

So _{
}

newnotion?

Of course these two examples don't prove anything, but they do give us ideas.

One idea is that it might be the case that
if a sequence (_{
)
} verconges to 0 then
it also converges to 0. This is in fact correct.

Exercise.

Prove that if a sequence _{
} verconges to 0 then
it also converges to 0.

The other idea comes from the feature of our first example that it is eventually zero, i.e.,

Perhaps all such sequences that are eventually zero verconge to zero? and Perhaps all sequences verconging to zero are eventually zero? It may not be 100% obvious, but this is in fact the case.

Proposition.

A sequence (_{
)
} verconges to 0
if and only if it is eventually zero.

**Proof.**

We must prove both directions. First, assume that _{
}
is eventually zero. That is, assume

We must show that _{
}<eventually zero

such that _{
=0
}

**Subproof.**

Hence _{
}<

For the other direction, assume that (_{
)
} verconges
to zero. We must show that _{
=0
}
_{
}<verconge

.)
Once we realise this the proof is straightforward.

So _{
=0
}