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I sometimes see this notation for convergence (speaking for functions): $f_n \to f$. And sometimes, I see following: $f_n \nearrow f$ or $f_n \searrow f$. What is the difference between $\to$ and other two?

Thanks.

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Please, try to make the title of your question more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. –  Julian Kuelshammer Nov 12 '12 at 21:04
    
@JulianKuelshammer thanks. It was hard to embed all information to the header, but I tried. I hope it is OK. Thanks. –  Mark Nov 12 '12 at 21:06
    
Now it's perfect. –  Julian Kuelshammer Nov 12 '12 at 21:07
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1 Answer

up vote 4 down vote accepted

The first generally means that the sequence of functions is pointwise non-decreasing, the second that it’s pointwise non-increasing. Sometimes a strict ordering is meant instead, so that the sequences are pointwise strictly increasing and pointwise strictly decreasing, respectively.

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OK, then $\to$ can mean one of these $\nearrow$ or $\searrow$? –  Mark Nov 12 '12 at 21:12
    
@John: Yes, but $\to$ also covers non-monotonic convergence. –  Brian M. Scott Nov 12 '12 at 21:13
    
Brian: now I got it. Thanks! –  Mark Nov 12 '12 at 21:15
    
@John: You’re welcome! –  Brian M. Scott Nov 12 '12 at 21:24
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