If (z_n) goes to p, then there is a real number M such that for every z_n,|z_n| less than or equal to M.
Is it true or false?
Please prove it.
True, I think.
Consider that z will at some point lie in a disc of radius r, no matter how small r may be.
To prove the given statement, we'll need to use the definition of convergence and the concept of boundedness.
Let's assume that the sequence (z_n) converges to p. This means that for every positive ε, there exists a positive integer N such that for all n ≥ N, the distance between z_n and p |z_n - p| < ε.
Now, let's consider the absolute value of each term in the sequence, |z_n|. We want to show that there exists a real number M such that for every term z_n, |z_n| ≤ M.
Since the sequence (z_n) converges to p, we know that there exists a positive integer N such that for all n ≥ N, |z_n - p | < 1.
Using the triangle inequality, we can write |z_n| = |z_n - p + p | ≤ |z_n - p | + |p|.
Since |z_n - p | < 1 (for n ≥ N), and |p| is a fixed value, we can choose M = max {|z_1|, |z_2|, ..., |z_N-1|, |z_N - p | + |p| }. This maximum of the absolute values of the first N-1 terms and |z_N - p | + |p| is finite, so we have found a real number M satisfying |z_n| ≤ M for every term z_n in the sequence.
Therefore, if the sequence (z_n) converges to p, there exists a real number M such that for every term z_n, |z_n| ≤ M.