## 3.1. Sequences

### Theorem 3.1.11: The Pinching Theorem

Suppose

*{a*and_{j}}*{c*are two convergent sequences such that_{j}}*lim a*. If a sequence_{j}= lim c_{j}= L*{b*has the property that_{j}}for alla_{j}b_{j}c_{j}

*j*, then the sequence*{b*converges and_{j}}*lim b*._{j}= L### Proof:

The statement of the theorem is easiest to memorize by looking at a diagram:*b*are between

_{j}*a*and

_{j}*c*, and since

_{j}*a*and

_{j}*c*converge to the same limit

_{j}*L*the

*b*have no choice but to also converge to

_{j}*L*.

Of course this is not a formal proof, so here we go: we want to show that
given any * > 0* there
exists an integer *N* such that
*| b _{j} - L | < *
if

*j > N*. We know that

Subtractinga_{j}b_{j}c_{j}

*L*from these inequalities gives:

But there exists an integera_{j}- L b_{j}- L c_{j}- L

*N*such that

_{1}*| a*or equivalently

_{j}- L | <and another integer- < a_{j}- L <

*N*such that

_{2}*| c*or equivalently

_{j}- L | <if- < c_{j}- L <

*j > max(N*. Taking these inequalities together we get:

_{1}, N_{2})But that means that- < a_{j}- L b_{j}- L c_{j}- L <

or equivalently- < b_{j}- L <

*| b*as long as

_{j}- L | <*j > max(N*. But that means that

_{1}, N_{2})*{b*converges to

_{j}}*L*, as required.