lambda's method? Mats' example code confuse me .

def memoize
  cache = {}
  lambda { |*args| 
    unless cache.has_key?(args)
      cache[args] = self[*args]
    end
    cache [args]
  }
end

factorial =  lambda {|x| return 1 if x== 0; x*factorial[x-1];}.memoize

puts factorial.call 10

The code is from book "The ruby programming language ". But it confuse me : how can the method(memoize) apply to a lambda as its method? Can lambda followed by other lambda with dot(.) as its own method?

lambda {|x| return 1 if x== 0; x*factorial[x-1];}.memoize

BTW: The above code works in irb, but ruby interpreter encounter error as following:

memoize.rb:11: private method `memoize' called for #<Proc:0x0000000103bba018@memoize.rb:11> (NoMethodError)

Why?