LimSup LimInf

Definition of limit supperior and limit inferior:

Let $(a_n)_{n \in \mathbb{N}}$ be a bounded sequence.

i) The limit superior of $(a_n)_{n \in \mathbb{N}}$ is defined by $\limsup_{n \to \infty} a_n= \lim_{k \to \infty} \left( \sup_{n \ge k} a_n \right).$

ii) The limit inferior of $(a_n)_{n \in \mathbb{N}}$ is defined by $\liminf_{n \to \infty} a_n= \lim_{k \to \infty} \left( \inf_{n \ge k} a_n \right).$

Characterization of the limit superior:

Let $(a_n)_{n \in \mathbb{N}}$ be a bounded sequence and let $a \in \mathbb{R}$ .
Then $\limsup_{n \to \infty} a_n = a,$ if and only if for every $\varepsilon > 0$ the following conditions are satisfied:

i) It holds that $a_n < a + \varepsilon$ for almost all (i.e., all except finitely many) $n \in \mathbb{N}$ .

ii) It holds that $a_n > a - \varepsilon$ for infinitely many $n \in \mathbb{N}$ .

How to use the applet

Use the slider $k$ to choose the tail $\{a_n : n\ge k\}$ .
Use the slider $\varepsilon$ to change the thickness of the $\varepsilon$ -neighborhoods around $\limsup a_n$ (pink) and $\liminf a_n$ (blue).
The red circle marks $\sup_{n\ge k} a_n$ (computed over the displayed tail).
The blue circle marks $\inf_{n\ge k} a_n$ (computed over the displayed tail).
The dashed horizontal lines are the levels $\limsup a_n$ (pink) and $\liminf a_n$ (blue).
The dashed bands show $\limsup a_n \pm \varepsilon$ (pink) and $\liminf a_n \pm \varepsilon$ (blue).
Points with $n<k$ are faded (they are “discarded” when looking at the tail).
Orange points are exceptions:
- above $\limsup a_n + \varepsilon$ , or
- below $\liminf a_n - \varepsilon$ . For a correct limit statement, you should be able to choose $k$ so that there are only finitely many such exceptions.
Pink points are “witnesses” that the sequence comes arbitrarily close from below to $\limsup a_n$ :

a_n > \limsup a_n - \varepsilon \quad \text{infinitely often.}

Blue points are “witnesses” that the sequence comes arbitrarily close from above to $\liminf a_n$ :

a_n < \liminf a_n + \varepsilon \quad \text{infinitely often.}

Suggested exploration

Fix a small $\varepsilon$ (e.g. $\varepsilon=1$ ) and increase $k$ .
Observe: orange points eventually disappear.
Keep $k$ large and shrink $\varepsilon$ .
Observe: you still get infinitely many pink/blue witness points near the corresponding dashed level.

Technical Description

Link to code.

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Theorem 2.16 page 28

For this example, I have selected the alternating sequence $a_n = (-1)^n \left( 10 + \frac{10 + 60\sin(0.8n)}{n} \right)$
to provide a clear distinction between $\liminf$ and $\limsup$ .

To clarify the first definition, the student can adjust the value of $k$ in $\sup_{n \ge k} a_n$ and $\inf_{n \ge k} a_n$ using the slider.

The current values of $\sup_{n \ge k} a_n$ and $\inf_{n \ge k} a_n$ are highlighted by circles with corresponding colors: red for the limsup and blue for the liminf.
In code tyheir are calculated as:

const getSupData = () => {
    const k_curr = Math.floor(kSlider.Value());
    let maxVal = -Infinity; let maxIndex = k_curr;
    for (let n = k_curr; n <= MAX_K; n++) {
        const val = values[n - 1];
        if (val >= maxVal) { maxVal = val; maxIndex = n; }
    }
    return { index: maxIndex, val: maxVal };
};

const getInfData = () => {
    const k_curr = Math.floor(kSlider.Value());
    let minVal = Infinity; let minIndex = k_curr;
    for (let n = k_curr; n <= MAX_K; n++) {
        const val = values[n - 1];
        if (val <= minVal) { minVal = val; minIndex = n; }
    }
    return { index: minIndex, val: minVal };
};

Both $\liminf$ and $\limsup$ are marked by dashed lines in their respective colors.

An illustration of the theorem is provided by allowing the student to move
the corresponding $\epsilon$ -slider for each of the limits.

For instance, in the case of the $\limsup$ , the points below $a+\varepsilon$ , are those finetely many $n$ to discharge and highlighted in orange, while the infinetely many points above $a - \varepsilon$ share the same limit color.

for (let n = 1; n <= MAX_K; n++) {
    board.create('point', [n, values[n - 1]], {
        ...DEFAULT_POINT_ATTRIBUTES,
        name: '',
        size: 1,
        strokeColor: COLORS.black,
        strokeWidth: 1,
        fixed: true,
        fillColor: () => {
            const k = kSlider.Value();
            const val = values[n - 1];

            if (n < k) return COLORS.lightGray;

            const epsSup = getEpsSup();
            const epsInf = getEpsInf();

            // "Almost all n are < Sup + eps" -> Exception if > Sup + eps
            if (val > TRUE_SUP + epsSup) return COLORS.orange;
            // "Almost all n are > Inf - eps" -> Exception if < Inf - eps
            if (val < TRUE_INF - epsInf) return COLORS.orange;

            // "Infinitely many n > Sup - eps" -> Witness if inside Sup tube
            if (val > TRUE_SUP - epsSup) return COLORS.pink;

            // "Infinitely many n < Inf + eps" -> Witness if inside Inf tube
            if (val < TRUE_INF + epsInf) return COLORS.blue;

            return COLORS.black;
        },
        strokeOpacity: () => n < kSlider.Value() ? 0.3 : 1
    });
}