$\limsup (a_n +b_n) = a + \limsup b_n$ when $a_n$ converges to $a$

Asked Oct 31 '20 at 02:26

Active Oct 31 '20 at 02:26

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I saw this property of $\limsup$ on wikipedia: https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior

If $a_n \rightarrow a$, then $\limsup (a_n + b_n) = a + \limsup b_n$.

I has been able to proved that if exists $a_N > a$, then this statement is true. However, I am struggling to show this statement is true when $a_n \leq a$ for all $n$.

asked Oct 31 '20 at 02:26

CuriousAlpaca

2

https://math.stackexchange.com/questions/337480/when-do-we-have-liminf-n-to-inftya-nb-n-liminf-n-to-inftya-n-liminf?rq=1 does this help? – PAM1499 Oct 31 '20 at 02:42
It does. Thanks. – CuriousAlpaca Oct 31 '20 at 03:01
You are welcome ;) – PAM1499 Oct 31 '20 at 03:02

$\limsup (a_n +b_n) = a + \limsup b_n$ when $a_n$ converges to $a$

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