Given income $y$ and a vector of commodity prices $p\in R_+^L$, the set of feasible consumption bundles is described by the budget correspondence, $B(p,y)=\{x\in R_+^L:px\leq y\}$. $B(p,y)$ is both upper and lower hemicontinuous, as proved in Chapter 8 Problem 2.2 in Mathematical Methods and Models for Economists by de la Fuente.
Here is the proof: To establish that $B$ is an lhc correspondence, we need to show that given any price-income seqwuence $\{(p_n,y_n)\}$ converging to $(p,y)>>0$ and an arbitrary point $x\in B(p,y)$, there exists a companion sequence of consumption bundles $\{x_n\}$ with $x_n\in B(p_n,y_n)$ for all $n$ that converges to $x$.
Let $x_n=x$ if $x\in B(p_n,y_n)$ and $x_n=\frac{y_n}{p_nx}x$ if otherwise. Notice that $x_n$ is feasible for $(p_n,y_n)$ by construction, because $x_n$ is defined as the largest fraction of the bundle $x$ that the consumer can afford with income $y_n$ and prices $p_n$. It's also clear that $\{x_n\}\to x$. If $x$ lies in the interior of the budget set, then we have $x_n=x$ for $n$ sufficiently large. Otherwise, $y=px$ and $\lim x_n=\lim\frac{y_n}{p_nx}x=\frac{y}{px}x=x$.
You can also see here for the proof.
But I'm having trouble understanding this proof. Here is the definition of lhc: A correspondence is lhc at $a$ if $\forall b\in F(a)$, $\exists a_n$ and $b_n$ such that $a_n\to a$ and $b_n\to b$.
But in this proof, $p_n$ is taken as an arbitrary sequence. $p_n$ should correspond to the $a_n$ sequence in the definition of lhc. In the definition, $a_n$ should be "there exists", not "for all". Can someone explain why?
