How does the red shift confirm that the universe is expanding?

According to Doppler's effect whenever there is a relative motion between a source of light and an observer, the apparent frequency of light received by observer, is different from the true frequency of light emitted by the source. When the distance between the source (galaxy) and observer is increasing, the apparent frequency is decreased and wavelength is increased, i.e., there is a shifting towards red end of the spectrum. Hence, we conclude that universe is expanding.