- Create a rigorous definition for the statement \(\lim\limits_{n \rightarrow \infty} x_n = \infty\). Use this definition to prove \(\lim\limits_{n \rightarrow \infty} \sqrt{n} = \infty\).
- What does your definition in (a) say about the particular sequence \((1,0,2,0,3,0,4,0,5,0)\)
Definitions of sequences and convergence: here, Definitions of subsequences: here.
Solution
(a) We say that
If and only if for every real number \(M \geq 0\), there exists an \(N \in \mathbf{N}\) such that for all \(n \geq N\), we have
Now, we want to show that
To do this, let \(M \geq 0\), we want to find an \(N\) such that for all \(n \geq N\), we have
Squaring both sides, we see that \(n\) needs to be greater than \(M^2\). Therefore, for every real number \(M > 0\), let \(N = \lceil M^2 \rceil + 1\). Then for for any \(n \geq N\), we must have \(\sqrt{n} > M\) and by definition, this implies that
as desired. \(\ \blacksquare\)
(b) Consider the sequence
Suppose that it converges to \(\infty\). Then by definition, every \(M \geq 0\), there exists an \(N\) such that for all \(n \geq N\)
But no matter how large \(N\) is, there will be infinitely many even values of \(n \geq N\) where \(a_n = 0\). So, the inequality \(a_n > M\) fails for those \(n\) values. Therefore, the definition is no longer is satisfied. Thus, this sequence doesn’t converge to \(\infty\).
References
Aleph 0