Brief explanation:

Given n elements (no duplicates),

• there are n! possible permutation sequences
• one of these possible sequences is a sorted sequence
• each comparision reduces number of possible sequences to be considered

For a given input,

• the algorithm follows a path from the root to a leaf
• requires one comparison at each level
• there are n! leaves for n elements (e.g. 3! leaves for 3 elements)
• height of such tree is at least log₂(n!),
  so number of comparisions required is at least log₂(n!)

log2(n!) = log2(1) + log2(2) + ... + log2(n/2) + ... + log2(n-1) + log2(n)
log2(n!) >=  log2(n/2) + ... + log2(n-1) + log2(n)
log2(n!) >=  (n/2)log2(n/2)  
log2(n!) =  Ω (n log2n)

Therefore, for any comparison-based sorting algorithm, the lower bound is Ω (n log₂ n) .