Assume $\lim_{n\rightarrow\infty}x_{n}$ exists. Prove that for any sequence $y_n$, we have
$$\limsup\{x_{n}+y_{n}\} =\lim_{n\rightarrow\infty}x_{n}+\limsup\{y_{n}\}$$
I got stuck on this question while revising and while once again intuitively this makes sense, this is the first time I'm dealin...
Happy Passover everybody,
Given a convergent sequence $x_{n}$ and bounded sequence $y_{n}$
I need to prove that $\limsup (x_{n}+y_{n})=\lim x_{n}+\limsup y_{n}$, when $n$ tends to $\infty$.
I chose $z_{n}=x_{n}+y_{n}$, we know that $z_{n}$ is bounded as being sum of two bounded sequences, so f...
The only difference I see that in the latter there is an assumption that $y_n$ is bounded, so we do not have to deal with the case $\limsup y_n=+\infty$.
Happy Passover everybody,
Given a convergent sequence $x_{n}$ and bounded sequence $y_{n}$
I need to prove that $\limsup (x_{n}+y_{n})=\lim x_{n}+\limsup y_{n}$, when $n$ tends to $\infty$.
I chose $z_{n}=x_{n}+y_{n}$, we know that $z_{n}$ is bounded as being sum of two bounded sequences, so f...
