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
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
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 .
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