# Limit superior and limit of a sequence

$$\{x_n\}$$ be a bounded sequence such that for every bounded function $$\{y_n\}$$,

$$\limsup\{x_{n}+y_{n}\} =\limsup\{x_{n}\}+\limsup\{y_{n}\}$$ then prove that {$$x_{n}$$} is convergent. This is a question asked in in real analysis exam. I started answering like this : If $$\limsup\{x_n\} = L$$ then for every $$\epsilon$$ we can find natural number N such that $$x_{n} for n $$\geq N$$ A similar result holds for given bounded sequence $$\{x_n\}$$ but I cannot say that $$L- \epsilon , $$n \geq N$$.

Can we prove this result from the same definition of limit ? Or should we use some other results which connects limit superior and limit of a sequence? What is the importance of boundedness of the sequence (is it given to say that limsup is not $$\infty$$ ?)

• LaTeX is quite happy to do brackets, e.g. \{ and \} give $\{ \text{ and } \}$ while if n is bigger than, or equal to, N then why not out n \ge N into LaTeX to give $n \ge N$. Moreover, how about \limsup to give $\limsup$? – Fly by Night Apr 28 '14 at 16:57

Choosing $y_n=-2x_n$ we conclude from $$\limsup_{n\to\infty} (x_n+y_n)=\limsup_{n\to\infty} x_n +\limsup_{n\to\infty} y_n$$ that $$\limsup_{n\to\infty} (-x_n)=\limsup_{n\to\infty} x_n +\limsup_{n\to\infty}(-2x_n)$$ that is $$-\liminf_{n\to\infty} x_n =\limsup_{n\to\infty} x_n -2\liminf_{n\to\infty} x_n$$ or equivalently $$\liminf_{n\to\infty} x_n=\limsup_{n\to\infty} x_n.$$ and we are done. $\qquad\square$
Remark: We used strongly the fact that the sequence $\{x_n\}$ is bounded to insure that both $\liminf_{n\to\infty} x_n$ and $\limsup_{n\to\infty} x_n$ and we can do arithmetic operations on them. Consider what happens when $x_n=(-1)^n n$.