Prove that limit(z -> z_0) z^2 = z_0^2
I must show that for any positive number E that there exists a positive number delta so that whenever this is true:
0 < |z - z_0| < delta
then this is true:
|z^2 - z_0^2| < E
To prove that the limit as z approaches z0 of z^2 is equal to z0^2, we need to show that for any positive number E, there exists a positive number delta such that whenever 0 < |z - z0| < delta, then |z^2 - z0^2| < E.
Let's consider the expression |z^2 - z0^2|. We can simplify this expression as follows:
|z^2 - z0^2| = |(z - z0)(z + z0)|
Since we want to find a delta such that |(z - z0)(z + z0)| < E, we can rewrite the expression as:
|(z - z0)(z + z0)| < E
Now, we can use the triangle inequality, which states that for any complex numbers a and b, |a + b| ≤ |a| + |b|.
Applying the triangle inequality to our expression, we have:
|(z - z0)(z + z0)| ≤ |z - z0| |z + z0|
Since we want to find a delta, we can assume that |z - z0| < 1 (or any other positive number that we choose). In that case, we can write:
|z + z0| = |z - z0 + 2z0| ≤ |z - z0| + |2z0|
Now, let's denote |2z0| as M (a positive number). Then we can write:
|z + z0| ≤ |z - z0| + M
Substituting this back into our expression, we have:
|(z - z0)(z + z0)| ≤ |z - z0| (|z - z0| + M)
Since we want this expression to be less than E, we can write:
|z - z0| (|z - z0| + M) < E
Now, assume that |z - z0| < 1 (or any other positive number that we choose). In that case, we can write:
|z - z0| + M < 1 + M
So our expression becomes:
|z - z0| (|z - z0| + M) < |z - z0| (1 + M)
Now, if we choose a positive number delta such that 0 < delta < 1, then we have:
0 < |z - z0| < delta < 1
Therefore, we can say that:
0 < |z - z0| < delta implies |z - z0| (1 + M) < delta (1 + M)
Since delta (1 + M) > 0, we can rewrite the expression as:
0 < |z - z0| < delta implies |z - z0| (1 + M) < delta (1 + M) < E
So, we have shown that for any positive number E, if we choose a positive number delta such that 0 < |z - z0| < delta, then |z^2 - z0^2| < E.
Therefore, from the definition of the limit, we can conclude that the limit as z approaches z0 of z^2 is equal to z0^2.