# Show that $\liminf(x_k \cdot y_k) = x \cdot \liminf(y_k)$ [closed]

We have a sequence $$(x_k)$$ in which $$\lim(x_k)= x$$ and $$x > 0$$. We also have that $$\exists k_0$$ such that $$y_k > 0 \space \forall k \geq k_0$$, then show that $$\liminf(x_k y_k) = x \liminf(y_k)$$.

I'm learning analysis and have this question, but I'm pretty lost on what to do. I think that we can tell that $$b_k$$ is bounded, but we don't have that it converges, so the Algebraic Limit Theorems don't apply.

• Welcome to Mathematics SE. Take a tour. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. Commented Jul 9 at 23:24
• just wondering, what do you mean by $liminf$ Do you mean $\lim \inf$ Commented Jul 9 at 23:25
• Can you show that if $A$ is the set of all partial limits of $(x_ky_k)_{k\in\mathbb{N}}$ and $\alpha \in A$ then $x\lim_{k\to\infty}inf(y_k) \leq \alpha$?
– X4J
Commented Jul 10 at 2:23
• Take advantage of the (actually unnecessary) hypothesis $y_k>0$: take the logarithms. Prove first that if a real sequence $(a_n)$ converges then for any real sequence $(b_n)$,$$\liminf(a_n+b_n)=\lim a_n+\liminf b_n.$$ Commented Jul 10 at 5:15
• Please edit your post to include some minimal work. To begin with, write down your preferred definition of $\liminf$. Commented Jul 10 at 5:22