# Mathematical Food for Thought

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

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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 .

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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.

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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.

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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 .