Time limited and Band limited Signal concepts

Time limited and Band limited Signal:

A time limited signal is one that is nonzero only for a finite length time interval.

A signal is said to be a band limited signal if all of it's frequency components are zero above a certain finite frequency.

Prove that , if a signal is time limited, it cannot be band limited and vice versa.

(Explained in the following video)

NOTE:

A signal can be simultaneously non time limited and non band limited.

Example:

Gaussian Function(Gaussian function is not a time limited signal and it's Fourier transform is also Gaussian function. So the signal is simultaneously non time limited and band limited.

(For derivation of Fourier Transform of Gaussian Signal check the following link:

http://www.cse.yorku.ca/~kosta/CompVis_Notes/fourier_transform_Gaussian.pdf)