This Instructable will go through, step by step, the general method for proving that a sequence converges to some limit via using the definition of convergence.
Quick definition of terms used in this Instructable:
1) Candidate: In a few steps we will look for a "candidate" for a limit of a sequence. What I mean by this is that we need to find a possible number that our sequence will converge to. It immediately seems unnecessary to prove convergence if we know that a sequence is convergent, but in analysis we are no longer concerned with "that" a sequence converges, but "why" it converges.
2) Epsilon: We use the Greek letter "Epsilon" to denote a very small distance of error between a sequence term and the limit.
Step 1: State the Sequence
Our sequence would be defined by some function based on the natural numbers in order for this procedure to work. For this Instructable I will use the sequence defined in the picture above.
Step 2: Find a Candidate for L
Before beginning our proof, we need to find a possible candidate for our limit. This is usually done using tricks from Calculus 2 which I omit from this Instructable. For this proof I claim our limit is 1.
Step 3: Let Epsilon Be Given
The first step in any convergence proof is to let our error range, Epsilon, be given.
Step 4: State Our "magic Number"
The magic number, k, tells us how far in the sequence we need to go before we end up inside our allotted error's distance of the limit. "k" will change depending on which specific Epsilon we pick.
Step 5: Look for Inequalities
Now begins the arduous process of creating a string of inequalities to show that we can make the distance between an element of our sequence and our supposed limit smaller than any error range, Epsilon. This will involve different tactics depending on the sequence, but in general for an introduction to Real Analysis the tools you'll want at your disposal include:
1) (Generalized) Triangle Inequality
2) Monotone Convergence Theorem
3) Reverse Triangle Inequality
4) Adding a special 0
5) Multiplying by a special 1
In the example above I multiplied 1 by n/n and then simplified by expression. Notice that I haven't dropped the absolute value bars yet.
Step 6: Drop the Absolute Value Bars If Possible
In the case of our example, n is always positive, therefore 1/n is always positive. So we can drop our absolute value bars.
Step 7: Define Our Magic K
Now that we have a simple expression which is greater than the distance from some element in our sequence and the supposed limit, we define our k. We can do this since k is up to us, it's not dependent on the sequence, it's dependent on Epsilon.
Note that we almost always want our k to be in the denominator of a fraction in order to use a special property in the next steps.
Step 8: State the Archimedian Property
The Archimedian property is almost always used when proving convergence from the definition, and we will apply it directly in the next step, however before we apply it we want to state it to be mathematically responsible. The statement is given in the image.
Step 9: Apply the Archimedian Property
Now that we've stated our Archimedian Property, and we have our absolute value, we want to apply it so that we know that the k-epsilon that we've found actually exists.
Step 10: Complete the Proof!
Now that we've found our k-epsilon, go back to step 4 and fill in what k-epsilon must be bigger than for the particular sequence you're working with. In this case our k-epsilon needs to be bigger than 1/Epsilon in order for our sequence to come within an Epsilon's distance from the limit, L.
Step 11: Instructional Video
In this video I go through a slightly more complicated example of proving that a sequence converges. I pick a slightly more complicated example to further discuss what each piece of the proof is and to go through finding those inequalities.