Using Laplace transforms I created a proof that cos(t) is not bounded above. I cannot find where the error is even though the proof doesn't use very complicated theorems. I am not so familiar with the limitations of Laplace transforms so I suspect that the error is at the start or the end where I use Laplace.

If cos(t) is not bounded above then for any c>1 there exists some t such that

Taking the Laplace transform of the equation

Assuming s is not equal to zero

There is no solution for real s, instead consider a purely complex value , where

Plotting gives the following graph:

As can be seen from the graph, the function is continuous for x>1

Now using the following limits:

and

Since then by the intermediate value theorem there exists some such that

Therefore a solution in the Laplace domain exists and this implies that a solution in the time domain exists for some

Is anyone able to spot the error in my reasoning? I didn't properly prove that the function was continuous when applying the intermediate value theorem but I don't think this is the blunder.