Chapter 15 of 24 - Metric Spaces and Continuity 2

Another brutal chapter. Brain was pushed to the limits with this one.

First chapter was "new metrics from old", about how sets and distances can be combined to form new metrics. Second was "the Cantor metric" (Cantor space, just being what is essentially a binary tree). Next was "equivalent metrics", that involved a lot of inequalities. Fourth was "spaces of functions" which covered sequences of functions and their limits (different from sequences and limits). Lastly was "the max metric on C[0,1]". Conceptually, this was a jump as C[0,1] is actually the set of all continuous functions that map from [0,1] to \mathbb{R}. Had some really tricky questions in that last section. The majority of my brain power went into understanding how to prove that the function you apply to functions when you integrate is continuous. Exhausted.

On to the next chapter.