Could you please explain me how to prove this property? This exercise is too hard for me to understand :(

I The following propertyI of the Hicksian demand him} = w rm: 1, ...,k I Let us now prove this result in the general case. I Suppose that the consumer can consume 1: different goods. I A consumption bundle is denoted by vector x = [11,...,Xk]. I Prices of the goods are given by vector p 2 {p1, ..., pk]. I The consumer's utility function is ufx]. Questions: 1. Suppose that the consumer wants to achieve a level of utility u at minimal expenditure. 1Write down the consumer's problem and derive the first order conditions. For the fr goods, just write one condition with respect to some good j {focus on the interior solutions]. . Solving this problem gives the optimal bundle x[p, u}, which can also be denoted by h{p, u], but this is just a matter of notation. For now, let's use x{p, u] . How can we then write the expenditure function e[p, u}? . Take the expression for efp, u] you have written in question 2 and differentiate both sides with respect to p,-. . Come back to the first order condition with respect to good j you derived in question 1. Use it to get an expression for pi. Use the expression for p,- to replace prices in the equation you derived in question 3. Now take the constraint of the consumer's problem, u[x] = u. Differentiate both sides with respect to p;. Use the result you get in question 5 in the equation of question 5. If you now use the notation l1,{p, u} for x;[p, u}, you should have the proof completed