{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0JARTXX4XP"
    }
  ],
  "body": "\n## Statement\n\nLet $(u_k)_{k \\ge 0}$ be a sequence of real numbers with $u_0 \\ge 0$, and let $(b_k)_{k \\ge 0}$ and $(c_k)_{k \\ge 0}$ be sequences of non-negative real numbers.\nAssume that $u_{k+1} \\le (1 + c_k)\\, u_k + b_k$ for each $k \\ge 0$.\nThen for each $n \\ge 0$,\n$$u_n \\le \\left( u_0 + \\sum_{k=0}^{n-1} b_k \\right) \\exp\\left( \\sum_{k=0}^{n-1} c_k \\right).$$\n\n## Remarks\n\nProof idea: by induction, $u_n \\le u_0 \\prod_{k\u003cn} (1 + c_k) + \\sum_{k\u003cn} b_k \\prod_{k\u003cj\u003cn} (1 + c_j)$. Then use $1 + c \\le e^{c}$.\nIt is the discrete analogue of [[Grönwall's inequality|07EJHQYR9A]], with sums in place of integrals.\nIt bounds the growth of errors in numerical methods such as the [[Euler method|0JARTXX4XP]].\n\n## Explanations\n\n### L1 human\n\nAt each step, the quantity grows at most by a factor $1 + c_k$ and an extra amount $b_k$.\nThe inequality says that the total growth is at most exponential in the sum of the rates.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "26D15",
      "39A22"
    ],
    "name": "Discrete Grönwall inequality",
    "role": "lemma",
    "sources": [
      {
        "locator": "Lemma 5.1 and (5.4)",
        "note": "S. Mischler, lecture notes \"Chapter 0, on the Gronwall lemma\", Section 5. Lemma 5.1 is the product form of the remark. With 1 + c \u003c= e^c it gives the statement, and (5.4) is the case of a constant c_k.",
        "url": "https://www.ceremade.dauphine.fr/~mischler/Enseignements/M2evol2018/chap0.pdf"
      },
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality"
      }
    ],
    "tag": "0NTM3J5MFE",
    "uses": [
      "07EJHQYR9A"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0NTM3J5MFE/",
  "links": [
    {
      "field": "uses",
      "target": "07EJHQYR9A"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0NTM3J5MFE.md",
  "msc": [
    "26D15",
    "39A22"
  ],
  "name": "Discrete Grönwall inequality",
  "prov": null,
  "role": "lemma",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0NT/0NTM3J5MFE.md",
  "status": null,
  "tag": "0NTM3J5MFE",
  "uri": "m2://0NTM3J5MFE",
  "url": "https://math2.tamnd.com/wiki/discrete-gronwall-inequality/",
  "writing": {
    "tag": "0NTM3J5MFE",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}