First part Second part Let $\alpha=\limsup|s_n|^{1/n}$ and $L=\limsup\left|\frac{s_{n+1}}{s_n}\right|$. We need to prove $\alpha\le L$. This is obvious if $L=+\infty$, so we assume $L<+\infty$. To prove $\alpha\le L$ it suffices to show $$\alpha\le L_1 \qquad\text{for any}\qquad L_1>L.$$ ...

I am struggling to digest and understand Theorem 12.2 (p. 76) in Elementary Analysis: The Theory of Calculus by Kenneth Ross Theorem 12.2 Let $s_n$ be any sequence of nonzero real numbers. Then we have lim sup $\mid$$s_n$$\mid$$^($$^1$$^/$$^n$$^)$ $\leq$ lim sup $\mid$$s_n$$_+$$_1...

Let $(a_n)$ and $b_n$ be bounded sequences of real numbers. Prove that $$\limsup _{n\to \infty}(a_n+b_n)\leq \limsup _{n\to \infty}a_n + \limsup _{n\to \infty}b_n$$ How can this be proved? Using the definition of limit, can I use the fact that $\sup|a_n + b_n|\leq \sup|a_n| + \sup|b_n|$?

I am stuck with the following problem. Prove that $$\limsup_{n \to \infty} (a_n+b_n) \le \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$$ I was thinking of using the triangle inequality saying $$|a_n + b_n| \le |a_n| + |b_n|$$ but the problem is not about absolute values of the seq...

The inequalities are: $$\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$$

$$ \limsup \left(f(h)+g(h)\right) \leq \limsup f(h)+ \limsup g(h).$$ How can we prove this? Any help would be appreciated.

I wish to prove the following two inequalities: Suppose $X$ is a subset in $\Bbb R$, and functions $f$ and $g$: $X\to \Bbb R$, and $x_{0}\in X$ is a limit point. Then: $$\lim\sup_{x\to x_0}(f(x)+g(x))\le \lim\sup_{x\to x_0}(f(x))+\lim\sup_{x\to x_0}(g(x))$$ and,$$\lim\inf_{x\to x_0}(f(x))+\...