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If $x_n$ converges to $x$, then $\lim \inf(x_n+y_n)=x+\lim \inf y_n$.

I know from a previous proof that $$\lim \inf(x_n+y_n)\geq \lim \inf(x_n) + \lim \inf(y_n).$$ From there I can do the following, since $x_n$ converges to $x$, $$\lim \inf(x_n+y_n)\geq x + \lim \inf(y_n).$$

But how do I get inequality?

Username Unknown
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1 Answers1

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Assumption: $\lim\inf(x_n+y_n)>x+\lim\inf(y_n)$. Then $$\begin{align*}\lim\inf(y_n)&<\lim\inf(x_n+y_n)-x\\ &=\lim\inf(x_n+y_n)+\lim\inf(-x_n)\\ &\leq \lim\inf(x_n+y_n-x_n)\\ &=\lim\inf(y_n),\end{align*}$$ (where in the 3rd line I used the inequality you already know from a previous proof) which means $\lim\inf(y_n)<\lim\inf(y_n)$. This is a contradiction!

So the assumption must be wrong which means that $\lim\inf(x_n+y_n)\leq x+\lim\inf (y_n)$. Since you already know that the same inequality holds with "$\geq$" instead of "$\leq$", both sides must in fact be equal.

Piwi
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