Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.
• Meta

Limitten. Topic: Real Analysis. November 23rd, 2006

Definition: A sequence has limit , where is a given real number, if for each there is an integer such that

for all integers .

——————–

Problem: (Stanford Math 51H) Prove that a sequence cannot have more than one limit.

Solution: This is logically true, as usual, but a rigorous argument is much more fun.

Suppose and . Letting , we know there exists such that

for

for .

So the above two inequalities are true for . Adding the two inequalities together, we get

.

However, by the triangle inequality we know (by setting is the usual one), so

.

Then

,

a contradiction. Hence can have at most one limit. QED.

——————–

Comment: The real definition of the limit is almost always complete overlooked in a regular calculus course (i.e. Calc AB and BC). But it’s pretty much the foundation of all of calculus so it is really quite nice to know and work with.

——————–

Practice Problem: (Stanford Math 51H, Sandwich Theorem) If are given convergent sequences with , and if is any sequence such that , prove that is convergent and .