Prove that budget constraint is Lower Hemi Continuos (LHC)
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 $x^n \in B(p_n,w_n)$, $x^n \not \rightarrow x$.