how do we know isotopes exist?

The mere fact that the atomic masses of most elements are not a whole number consitutes an evidence that isotopes exist.

The atomic mass of an atom is the sum of protons and neutrons. Given these are integer numbers the atomic mass of an atom is also an integer number. Then why there are so many elements whos atomic mass is not a whole number?

Because the atomic mass of an element is the wieghted average of the masses of the different atoms, so the fact that you obtain a decimal number is because there are atoms with one weight and atoms with different weight. Given that an elements is identified by the number of protons (all the atoms of a given element have the same number of protons) the only possible difference is in the number of neutrons. An these are isotopes: atoms of the same element (same number of protons) with different number of neutrons.

We know isotopes exist since we "see" them isolate utilizing a mass spectrometer. On the off chance that lone 1 sort of component existed, when we play out this trial, we'd see one major heap of molecules all being diverted by a similar sum. Since the masses of isotopes are all unique, they are altogether diverted by somewhat extraordinary sums, making diverse "piles" of the isotopes that component has.