Mathematical Presentation:
Consider the utility function U = U(x_{1}, x_{2}). Differentiating totally, we get the following: dU = U_{1}dx_{1} + U_{2}dx_{2} = 0 (as along the indifference curve utility is constant, dU = 0). Therefore, , which is the slope of the indifference curve. It is negative since U_{1}(x_{1}, x_{2}) >0 and U_{2}(x_{1}, x_{2}) >0 by assumption of non-satiation of all goods. Thus, indifference curve is downward sloping because all goods are non-satiated, choice is continuous, reflexive and complete. Economic meaning All goods are non-satiated i.e., larger (lower) consumption leads to lager (lower) utility. Hence, for given x_{2}, as x_{1 }increases, utility increases. Thus, to maintain same level, utility must be reduced, which is possible by reducing x_{2}. Hence, as x_{1} increases, x_{2} must decreases in order to maintain same level of utility. That is why indifference curve is downward sloping.