Prove that budget constraint is Lower Hemi Continuos (LHC) I need to prove that the following constraint is LHC.
$B=\{x \in R^n : px\leqslant pw)$
But Im not capable of finding and sequence $\{x_n\}$ such that $x_n \in B(p_n,w_n) \forall n$ and that $x_n\longrightarrow x$.
I tried with setting  $x_n=\frac{x}{1+\beta^n}$, $p_n=\frac{p}{1+\beta^n}$ , $w_n=\frac{w}{1+\beta^n}$ because in that case $x_n\longrightarrow x$. and $x_n \in B(p_n,w_n) \forall n$ but I thing I'm not covering the $\forall p_n, w_n$ part.
Help please and thanks in advance.
I don't believe it is lower semicontinous.
Let $w = (0,\dots,0)$, $p \in \mathbb{R}^n_+$ be any vector such that $p_1 = 0$ (the first coordinate being 0).
The allocation $x=(1,0,\dots,0) \in B(p,w)$.
Define the sequence $p_n = p  + (\frac{1}{n},0,\dots,0)$ and $w_n = (\frac{1}{n},0,\dots,0)$. $w_n \rightarrow w$ and $p_n \rightarrow p$.
For any $x^n \in B(p_n,w_n)$, $p_n x^n_1 \leq w_np_n$, so $x_1^n \leq \frac{1}{n}$.
Hence for any sequence such that $x^n \in B(p_n,w_n)$, $x^n \not \rightarrow x$.