Let $(x_n)_n$ and $(y_n)_n$ be sequences in $\mathbb{R}$, such that $(x_n)_n$ converges to $x \in \mathbb{R}^+$. Suppose that $(y_n)_n$ is bounded.
Prove that $$\limsup_{n \to +\infty} x_n \cdot y_n = x \cdot \limsup_{n \to +\infty} y_n$$
I know that $x=\limsup_{n \to +\infty} x_n$, so it basically comes down to proving $\limsup_{n \to +\infty} x_n \cdot y_n = \limsup_{n \to +\infty} x_n \cdot \limsup_{n \to +\infty} y_n$, but that's where I'm stuck...
You can not prove your lemma because it is wrong. Let $(x_n)=(1,2,1,2,1,2,...)$ and $(y_n)=(2,1,2,1,2,1,...)$ then their product is constant $2$, the product of the lim sup is $4$.
Given an $0<ε<x/2$ there is an $N$ so that for all indices $n\ge N$ $$ x-ε\le x_n\le x+ε\text{ and } y_n\le \limsup_{k\to\infty} y_k+ε. $$ Additionally, for infinitely many of these indices $n$ one has $$ \limsup_{k\to\infty} y_k-ε\le y_n. $$ This can be combined to prove the claim.