I have made a solution for the following question and I'm wondering if it's correct. I think that something is missing here. Can you help me complete the solution?
Let $f$ be a function. The fuction $f$ is monotonic increasing and it is not bounded.
Prove that $$\lim_{x\to\infty} f(x)=\infty$$
Solution:
Assume in contradiction that $$\lim_{x\to\infty} f(x)=L$$
Therefore there is some $M\in \Bbb R$ such that $\left|f(x)-L\right|<\epsilon$.
Therefore, $L-\epsilon<f(x)<L+\epsilon$
We can now say the f(x) is bounded, which is a contradiction.
Something is missing here. I know that in sequenses, every convergent sequence is bounded. Is there a theorem for functions also?
Thanks,
Alan
Best Answer
Let $M > 0$ be any given positive real number, since $f$ is unbounded, there is a natural number $N$ such that $f(N) > M$. Thus if $x > N \to f(x) > f(N) > M$. This shows $\displaystyle \lim_{x\to \infty} f(x) = +\infty$