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I know that $\limsup (x_n)+\limsup (y_n)\ge \limsup (x_n+y_n)$, but I'm not sure how to show that $\liminf (x_n)+\liminf (y_n)\le \liminf (x_n+y_n)$. Please help me.

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    Do you know how to prove that $\limsup(x_n+y_n) \le \limsup(x_n) + \limsup(y_n)$? – 2017-02-06
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    Hint: $\limsup (-x_n) = - \liminf x_n$ – 2017-02-06
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    Yes I know how it prove.....but please independent proof – 2017-02-06

3 Answers 3

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Fix $n$ for the moment and let $l \ge n$. Since $\inf_{ k \ge n} x_k \le x_l$ and $\inf_{ k \ge n} y_k \le y_l$, we see that $\inf_{ k \ge n} x_k + \inf_{ k \ge n} y_k \le x_l + y_l$. Hence $\inf_{ k \ge n} x_k + \inf_{ k \ge n} y_k \le \inf_{ k \ge n} ( x_k + y_k)$.

Since these are non decreasing sequences, we have $\inf_{ k \ge n} x_k + \inf_{ k \ge n} y_k \le \liminf_{ k } ( x_k + y_k)$ and hence $\liminf_{ k } x_k + \liminf_{ k } y_k \le \liminf_{ k } ( x_k + y_k)$.

4

Start with the inequality for $\limsup$ that you know, and use the hint of RRL. \begin{align} \limsup (-x_n) + \limsup (-y_n) &\ge \limsup (-x_n-y_n)\\ -\liminf x_n - \liminf y_n &\ge -\liminf (x_n+y_n)\\ \liminf x_n + \liminf y_n &\le \liminf (x_n+y_n) \end{align}

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    Please independent proof – 2017-02-06
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Recall that $\liminf x_n = \lim_{n \to \infty} \inf_{k \geqslant n} x_k.$

Note that for all $k \geqslant n$

$$x_k \geqslant \inf_{j \geqslant n} x_j,\\ y_k \geqslant \inf_{j \geqslant n} y_j$$

Thus,

$$x_k + y_k \geqslant \inf_{j \geqslant n} x_j + \inf_{j \geqslant n} y_j \\ \implies \inf_{k \geqslant n} (x_k + y_k) \geqslant \inf_{j \geqslant n} x_j + \inf_{j \geqslant n} y_j.$$

Now finish by taking the limit of both sides as $ n \to \infty.$