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Real Analysis 7 | Understanding Cauchy Sequences and Completeness
Learn about Cauchy sequences and completeness in Real Analysis. Understand the concept of sequences getting arbitrarily close without needing to know the limit. Explore the idea of convergence without specifying a particular number in the definition.
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Video Transcript
Hello and welcome back to Real Analysis.
and as always i want to thank all the nice people that support this channel on Steady or Paypal.
Todays part 7 is about Cauchy sequences and completeness.
For starting this topic lets recall that we already considered sequences with a special property.
Namely convergent sequences.
Which means there is a number a such that the sequence members here
get arbitrarily close to this number a eventually.
You already know the formal way to say this, which is for all epsilon there exists a capital N
such that for all indices n greater than this N
the distance between a and aN is less than the given epsilon.
Now the problem with this definition is that you need to know the limit to show convergence.
Simply because we measure the distance to this a.
Hence there is a different idea or a different property a sequence can have, which does not need such an argument.
number a in the definition.
For this lets look at the number line again and at a sequence which should converge.
So here we have a1, a2, a3 and so on.
And the sequence members accumulate here, so there should be a limit here.
However we don't want to use this limit to describe what happens here.
Indeed what happens here is that the sequence members themselves get closer and closer to each other.
Hence what we want is that the sequence members lie arbitrarily close to each other eventually.
So everything is about the distance you can measure between two sequence members here.
Which is the absolute value of aN minus aM.
and then this should be less than epsilon we choose at the beginning.
Therefore the formal way then reads for all epsilon greater 0
we find a capital N
such that for all indices called n and m afterwards
we have that the distance between the sequence members is
less than epsilon.
And now a sequence with this property we call a Cauchy sequence.
Ok, so lets put that into a definition.
This is exactly the definition you might have already seen in my start learning reals series.
There i also showed you the important fact that for a sequence of real numbers
we have that being a Cauchy sequence is equivalent to being a convergent sequence.
The proof of this direction you can see in part 2 of the course.
and the other one is exactly the completeness axiom.
So the completeness axiom tells us that there are no holes in our complete real number line.
Now what you really should remember is when we work in the real numbers
we don't have to distinguish Cauchy sequences and convergent sequences.
They have different definitions, but for the real numbers they mean the same thing.
Therefore we are able to use the one or the other definition.
depending what is useful in our context.
and soon we will see that the definition of the Cauchy sequences make a lot of things easier.
However before we apply this in examples lets discuss another important property.
It's called dedekind completeness and a property for subsets of the real numbers.
If m is such a set and also bounded from above, then we know the supremum exists.
So there is a least upper bound as a number in R.
Please recall that we defined the supremum in the last video.
and maybe not so surprising we have the same thing for the infimum as well.
So if we have a set that is bounded from below, then the infimum exists as a real number.
Ok, let me explain how we can prove this statement.
