Proving Divergence of (tn + sn)

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The discussion focuses on proving the divergence of the sequence (tn + sn) where (tn) diverges and (sn) converges. The user explores two cases: one where (tn) is unbounded and another where (tn) is bounded and oscillatory. In the unbounded case, it is established that (sn + tn) diverges due to the unbounded nature of (tn). In the bounded case, the user grapples with the implications of convergent subsequences of (tn) having multiple limits, questioning whether this leads to (sn + tn) also lacking a unique limit. The discussion hints at a potential contradiction proof involving the triangle inequality.

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I have a super round about way to prove this, but I'm having trouble finding a succinct proof

Let (tn) be diverge and (sn) converge. Show (tn+sn) diverges

The way I was doing involved considering that tn was unbounded, then showing it (sn+tn) is divergent.

Then I had to consider that tn is bounded and oscillatory, consider convergent subsequences, and show (sn+tn) had no unique limit, and therefore diverges. This part of the proof seemed less clear and I'm not sure if I can assert that because I have convergent subsequences of (tn) with multiple limits that (sn+tn) also has multiple limits.

I figure there has got to be some simple contradiction proof involving some triangle inequality trick that I'm just missing.

Thanks
 
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Hint: the sum (or difference) of two convergent sequences is convergent.
 
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