Recall that a sequence (_{
)
} converges to
This page gives three examples of convergent sequences, all properly proved. The exercises for this page give further examples for you to look at.
Consider first the constant sequence where each term is _{ = }.
Proposition.
(_{
)
} converges to
Proof.
We must prove that
Remark.
For this particular proof, the statement starts
with a
Subproof.
Hence
Now consider the sequence (_{ ) } with terms _{ =+1/ }.
Proposition.
(_{
)
} converges to
Proof.
Remark.
Again we must start by letting
Subproof.
Let
Subproof.
Let
So 1+1/
Subproof.
Let
Subproof.
Assume
Then
So
Hence (
Hence
Hence
Hence
Let
Bernoulli's inequality.
Let
Proof.
By induction on
Theorem.
If 0
Proof.
Remark.
We have to estimate
Subproof.
Let
Let
Let
Subproof.
Let
Then
Thus
So
So
So
The cases for other values of
The cases of other values of
Now consider the sequence (
Proposition.
(
Proof.
Remark.
We must start by letting
Now we are ready to give the proof.
Subproof.
Let
Subproof.
Let
So
Subproof.
Let
Subproof.
Assume
Then
So
So
Hence (
Hence
Hence
Hence
Proofs like this should not be so difficult. There is always an algebraic calculation to make, and it might help to do this in rough first. The proof itself follows the format of the proof rules exactly, and once the rough calculation has been done can be written out almost without any thinking at all.
Exercise.
Show, using similar methods, that the
sequence defined by