A lesson taught by lim sup

The concept of lim sup's (for sequences) taught me a valuable lesson today.

Intuitively, lim sup x_n is the largest value to which the sequence can possibly converge to (formal definition in Wikipedia). Now, lim sup's are very useful things, because every (real?) sequence has one, even those that are divergent. However, because a lim sup isn't actually a limit, we can't add them or multiply them and expect them to behave like plain old limits (we can multiply them by a scalar though); for example:

Take x_n = 0 if n odd, 1 if n even (i.e. 0, 1, 0, 1, ...). Obviously, lim sup x_n = 1
Take y_n = 1 if n odd, 0 if n even (i.e. 1, 0, 1, 0, ...). Obviously, lim sup y_n = 1
But, lim sup(x_n + y_n) = lim sup 1 = 1, not (lim sup x_n) + (lim sup y_n) = 1 + 1 = 2. Similarly, lim sup [(x_n)(y_n)] = lim sup 0 = 0, not (lim sup x_n)(lim sup y_n) = 1.

Unfortunately...we like these properties (ok, *I* like these properties). It makes life alot easier. Lacking them, I got stuck on an assignment question, resulting in a voyage to the calculus professor's office to beg for help/hints. He gave me the following:

Lemma (is this important enough to be a Theorem?):
If lim sup x_n = p, and lim y_n (as n tends to infinity) = q > 0*, then lim sup [(x_n)(y_n)] = pq.

Proof:
It has been shown that if lim sup x_n = p, then for every epsilon > 0:
There exists an integer K such that x_n < p + epsilon when n > K
There exists a subsequence n_i such that x_(n_i) > p - epsilon for all n_i.
Also, there exists a L such that q - epsilon < y_n < q + epsilon when n > L.

Then, taking M = max{K, L}:
(x_n)(y_n) < (p + epsilon)(q + epsilon) = pq + (p+q)epsilon + (epsilon)^2 for all n > M
[(x_(n_i)][y_(n_i)] > (p - epsilon)(q - epsilon) = pq - (p+q)epsilon + (epsilon)^2 for all n_i > M

Now (+/-)(p+q)epsilon + (epsilon)^2 = epsilon', and we can make epsilon' > 0 to be as small as we want by tweaking epsilon.

So for all epsilon' > 0, there exists M and and subsequence such that
(x_n)(y_n) < pq + epsilon' for all n > M
[x_(n_i)][(y_(n_i)] > pq - epsilon' for all n_i

Lesson learnt:
Prove only what we actually need, not what delusions of grandeur tells us we need. This lemma made me feel extremely guilty afterwards since the proof is quite straightforward, and I should have thought of needing the lemma (instead of trying to convince oneself that lim sups are preserved under multiplication "in general" even if they clearly aren't).

In retrospect: (realizing this as I type it up) I don't feel as comfortable with the proof any more...not certain of the line "Now (+/-)(p+q)epsilon + (epsilon)^2 = epsilon', and we can make epsilon' > 0 to be as small as we want by tweaking epsilon." However, since epsilon' is a quadratic function of epsilon, and a parabola takes on all positive real values, we should be ok...I hope.

Onto something else:
Orchestra rehearsal (both chamber and normal) did not go well today...don't feel like blogging about it though. Lesson learnt from the rehearsal is - check emails. Often. Not merely after midnight, when the information in the email is no longer relevant because the rehearsal is over. Thanks, email, for telling me that we are bringing in a new timpano, and that I'll need an extension cord with an hour to set up (another instrument) AFTER the rehearsal.

A list of things to do (as usual; some are repeated from previous blogs because well, they aren't done yet):

• Chemistry lab. Plogging away...calculating pH of acetic acid titrated with sodium hydroxide...*yawn* ok, ok, working on it. Still don't understand why CH₃COOH is every first-year instructor's favourite example of a weak acid. It is so typical...
  
• CS assignment - it has more math than coding in it though, which is a nice change (working with strong induction).
  
• Physics assignment. I am sorry for forgetting about you. Kind of. Actually, to be honest, not really. But the due date is coming. (Also: check schedule for date of physics final)
  
• Thermodynamics assignment. Due on the same day as Physics.
• Algebra assignment. It looks interesting.
• Calculus bonus problems. Along with the assignment that will be issued tomorrow.
  
• Statistics (now that I am mostly caught up, I would like to stay caught up).
• That professional development thing (just found out that it's due next Friday not the current one...so more procrastination is in progress).
• Read Vogel. His inorganic text anyway.
  
• Somehow borrow an extension cord since nobody seems to have one (I'd rather not buy it since I'll probably never use it again after the performance)