Why is the wavelength lesser if the frequency is higher for any wave?
For any wave, the wavelength W, frequency F and phase velocity of the wave V is related by F = V/W. When different waves of the same kind are compared it is the velocity of the waves that is usually constant. For example all electromagnetic waves have the same velocity, different types of sound waves in a medium have the same velocity.
The frequency of the waves is inversely proportional to the wavelength. This is required as F*W has to remain the same, if there is an increase in the frequency there has to be a corresponding decrease in the wavelength and vice versa.
This is the reason why when we compare two waves of the same type, the wave with the smaller wavelength has a higher frequency.