It appears I'll need some q-tips and rubbing alcohol for my keyboard. This section nailed the substance of it:

Quote:


Start with a FINITE interval that contains all the "an". Since the sequence is bounded,( -L, L )cut it into halves. At least one half must contain "an" for infinitely many n. and then continue with that half and cut it into two halves, so on and so forth. This process constructs a sequence of intervals where the common element is the limit of a subsequence.




Excellent work, Padova! (Admittedly, I only took real analysis for half a quarter and had to drop to attend to a family health crisis. Previously, I took complex analysis also and did well.)