input: from typing import List
  
  def cover_points_with_min_rectangles(points: List[List[int]], width: int) -> int:
      """
      Given a 2D list of integers points, where each element points[i] = [x_i, y_i], and an integer width, 
      your task is to cover all the points using rectangles.
  
      Each rectangle must have its bottom boundary at (x_1, 0) and top boundary at (x_2, y_2) 
      such that x_1 <= x_2, y_2 >= 0, and the width constraint x_2 - x_1 <= width must hold.
  
      Any point is considered covered if it lies within or on the boundary of a rectangle.
  
      Return the minimum number of rectangles required to ensure every point is covered by at least one rectangle.
  
      Note: A point can be covered by multiple rectangles.
  
      Examples:
      >>> points = [[2,1],[1,0],[1,4],[1,8],[3,5],[4,6]], width = 1
      2
      >>> points = [[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6]], width = 2
      3
      >>> points = [[2,3],[1,2]], width = 0
      2
  
      Constraints:
      1 <= len(points) <= 10^5
      points[i].length == 2
      0 <= x_i == points[i][0] <= 10^9
      0 <= y_i == points[i][1] <= 10^9
      0 <= width <= 10^9
      All (x_i, y_i) pairs are unique.
      """
output:     def cover_points_with_min_rectangles(points: List[List[int]], width: int) -> int:
      # Sort points by their x-coordinates
      points.sort()
      
      num_rectangles = 0
      i = 0
      n = len(points)
  
      while i < n:
          num_rectangles += 1
          current_end_x = points[i][0] + width
          # Move to the next point that is not covered by the current rectangle
          while i < n and points[i][0] <= current_end_x:
              i += 1
  
      return num_rectangles