1. Working from definitions directly, prove that a Cauchy sequence with a convergent subsequence must be convergent. 2. Prove that if we have two series of positive terms and we know that one is convergent and that each term of the other is bounded by a constant multiple of the corresponding term of the first, then the other series is also convergent. 3. There is a simpler version of the Ratio Test for series with positive terms that will be sufficient for our purposes. Namely we assume also that the ratios of successive terms form a convergent sequence. The Ratio Test for such series then says that if the limit of this sequence is less than 1 then the series is convergent whereas if the limit is greater than 1 then the series is divergent. If the limit is 1 then the Ratio Test is inconclusive. Prove this using the fact that geometric series with positive terms and common ratio less than 1 are convergent. [If you find this too hard, try to make progress with some examples.]

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