Existence of Equilibrium Prices:

Existence of a set of equilibrium (relative) prices is tried to be seen by looking  into the conditions  that ensure the following:  

Acertain that there exists at least one set of prices for which, 

a)  individuals maximise their utilities subject to their budget constraints;  and
b)  the demands are met  by  the existing stocks of the  commodity  in  the hands of the consumers.

The  idea then  is to start with  some initial arbitrary set of prices at which the  economy is not  in  equilibrium. Some commodities may be in excess  supply  while others  in  excess demand. Some other market may be clearing at  that  price  vector. Holding  the other (n-1)  prices  constant,  find  the  equilibrium  price in  the market for good I. Term this as provisional equilibrium price  4'. 

Keeping  4' and the other (n-2) prices constant, solve for the equilibrium  price  in the market good 2. Call this price  P₁' .  By  changing P₂ from its initial position to P₁', you have disturbed the  initial equilibrium price of market I. Since the  system of equations is Simultaneous,  it  is bound  to happen. Using the provisional prices,  4'  and  Pi,  solve per a provisional  P,'  and  the proof proceeds until a complete set of provisional relative prices  has  been calculated.

The phase of second interaction  starts  by  holding  P₂'......P_n'  constant  and calculating the new  equilibrium price  (P₁')  for the first good. Proceeding the same manner,  we get  an  entire  new  set  of  provisional  relative prices (P₁',p₂'  ....  P_n).  The iteration goes on until  a reasonable approximation to a set of equilibrium prices is achieve.