Let $x_n$ be a sequence of non-negative real numbers such that $\forall n, x_{n+1}\leq x_n+ \frac{1}{n^2}$ Prove that $x_n$ converges.  The problem is trivial whenever $x_{n}$ is an increasing sequence. In the general case I managed to prove that $(x_n)$ is bounded. It remains to prove that $(x_n)$ has only one accumulation point, but that seems difficult. Any hint is welcome.
Let $x_n$ be a sequence of the type described above. It is not monotonic in general, so boundedness won't help. So, it seems as if I should show it's Cauchy. A wrong way to do this would be as follows (I'm on a mobile device, so I can't type absolute values. Bear with me.) $$\left|x_{n+1} - x_n\right| \leq \frac{1}{n^2}.$$ So, we have $$ \left|x_m -x_n\right| \leq \sum_{k=n}^{m} \frac{1}{k^2}$$ which is itself Cauchy, etc., etc. But, of course, I can't just use absolute values like that. One thing I have shown is that $x_n$ is bounded. Inductively, one may show $$ \limsup_{n \to \infty} x_n \leq x_k + \sum_{k=n}^{\infty} \frac{1}{k^2},$$ although I'm not sure this helps or matters at all. Thanks in advance.  Disclaimer I've noticed that asking a large number of questions in quick succession on this site is often frowned upon, especially when little or no effort has been given by the asker. However, I am preparing for a large test in a few days and will be sifting through dozens of problems. Therefore, I may post a couple a day. I will only do so when I have made some initial, meaningful progress. Thanks.
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Relevance  Overview of current tags (aside from the discussion tag) for the first ten  (           ) questions relating to  (ordered by relevance)                              1   2   3   4   5   6   7   8   9   10     moderators              x           x   x   x     community               x           x   x     etiquette                   x     users                       x     company                     x     moderation                      x     code-of-conduct                     x       x       x     deleted-questions                               x     legal                                                   x   These tags are obviously not the same while it is arguably the same topic (it is even for the same, specific, search term) .  Instead of using multiple searches one might wish to control tracking these question by adding some single relevant tag to watched tags or ignore tags. But, this is difficult when they do not use the same tag. 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