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1. Let a_n be a sequence. If a_n -> b and a_n -> c then b = c.

Is this fact useful? I mean, are there any proofs that two things are equal by showing that there exists a sequence which approaches both of them?
2. The thing with analysis topics like this is that although they might be fairly obvious and not very useful, if they weren't true it would be very not-useful, so you have to prove mathematically that they're true (because intuition isn't good enough). Uniqueness of a limit isn't necessarily obvious, hence the requirement that this be proved.

It's probably more useful in cases where you're given, say, that (with unknown), but if you know what the limit of the sequence is then you can work out by just setting it equal to the limit.

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Updated: January 21, 2010
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