Mathematical Presentation:

Consider the utility function U = U(x₁, x₂). Differentiating totally, we get the following: dU = U₁dx₁ + U₂dx₂ = 0 (as along the indifference curve utility is constant, dU = 0). Therefore, , which is the slope of the indifference curve. It is negative since U₁(x₁, x₂) >0 and U₂(x₁, x₂) >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 x₂, as x₁ increases, utility increases. Thus, to maintain same level, utility must be reduced, which is possible by reducing x₂. Hence, as x₁ increases, x₂ must decreases in order to maintain same level of utility. That is why indifference curve is downward sloping.