Proofs using $\liminf$ and $\limsup$.

Question

Let {$x_n$} and {$y_n$} be two bounded sequences.

Suppose that {$x_n$} converges to x. Prove that

$\liminf_{n\to\infty}$ ($x_n$ + $y_n$) = x + $\liminf_{n\to\infty}$ $y_n$ and

$\limsup_{n\to\infty}$ ($x_n$ + $y_n$) = x + $\limsup_{n\to\infty}$ $y_n$.

I know that both $\liminf_{n\to\infty}$ (c + $x_n$) = c + $\liminf_{n\to\infty}$ $x_n$ and

$\limsup_{n\to\infty}$ (c + $x_n$) = c + $\limsup_{n\to\infty}$ $x_n$.

Can you treat the sequence {$x_n$} as just the value of x in the proof since we already know it converges to x?

Answer 1

No, you can't treat the sequence as its limit. Intuitively, it could be that the difference between $x_n$ and $x$ might be large enough to "throw off" the $\liminf$ or $\limsup$.

The better approach is to work from the definitions. To say $\liminf_{n\to\infty}(x_n + y_n) = z$ is to say that for any $\epsilon > 0$, there is an $N$ so that $|\inf\{x_n + y_n : n > N\} - z| < \epsilon$; so that's what we need to show. Say $\liminf_{n\to\infty}y_n = y$. Then we have $|\inf\{y_n : n > N\} - z| < \epsilon/2$ for some large enough $N$. Since $\lim_{n\to\infty}x_n = x$, we have that $|x_n - x| < \epsilon/2$ for all $n > N_1$ for some $N_1 > N$. Try to show that $|\inf\{x_n + y_n : n > N_1\} - (x + y)| < \epsilon$.