3.1. Sequences
Theorem 3.1.11: The Pinching Theorem
Suppose {aj} and {cj} are
two convergent sequences such that
lim aj = lim cj = L. If a sequence
{bj} has the property that
aj bj cjfor all j, then the sequence {bj} converges and lim bj = L.
Proof:
The statement of the theorem is easiest to memorize by looking at a diagram: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 | bj - L | < if j > N. We know that
aj bj cjSubtracting L from these inequalities gives:
aj - L bj - L cj - LBut there exists an integer N1 such that | aj - L | < or equivalently
- < aj - L <and another integer N2 such that | cj - L | < or equivalently
- < cj - L <if j > max(N1, N2). Taking these inequalities together we get:
- < aj - L bj - L cj - L <But that means that
- < bj - L <or equivalently | bj - L | < as long as j > max(N1, N2). But that means that {bj} converges to L, as required.