This episode is concerned with another invariance property continuous functions have. After having introduced and exemplified sequential compactness, we provide some intuition behind it. Then we prove that images of sequentially compact spaces under continuous maps are themselves sequentially compact. The immediate application to the particular case of functions mapping into the real numbers shows that continuous real-valued functions defined on sequentially compact spaces admit their supremum and infimum; that is, the maximum and the minimum of the image exists.

Picture: William Murphy from Dublin, Ireland, CC BY-SA 2.0, via Wikimedia Commons