Given function $ f : I \to \Bbb R$, if it is strictly increasing, bounded and continuous, then $I$ must be a bounded interval

Question

Can someone help explain why this statement is false

Answer 1

You use the fact that "strictly increasing continuous functions send sequences which diverge to infinity to sequences which diverge to infinity". This is false, and overall your claim is false. For instance, consider the function $$f(x)=\arctan x$$

Answer 2

The statement you are trying to prove is not true. Consider, for example, the function $\arctan$.

Answer 3

Or the poor man's $\arctan x:$ $f(x) = 1 - 1/x$ on $[1,\infty).$