We continue to live on through others. We see this in a few ways.
An important part of the seder expereince is meditating on this in its many forms.
ת"ח, כָּל מַאן דְּאִשְׁתְּדַל בְּאוֹרַיְיתָא, אָחִיד בְּאִילָנָא דְּחַיֵּי. כֵּיוָן דְּאָחִיד בֵּיהּ, בְּכֹלָּא אָחִיד, דִּכְתִּיב עֵץ חַיִּים הִיא לַמַּחֲזִיקִים בָּהּ וְגוֹ.' רִבִּי יִצְחָק אָמַר, כָּל מַאן דְּיִשְׁתְּדַּל בְּאוֹרַיְיתָא, חֵירוּ אִית לֵיהּ מִכֹּלָּא, חֵירוּ מִמִּיתָה, כְּמָה דְּאַמָרָן. בְּגִין דְּחֵירוּ עָלֵיהּ שַׁרְיָא, וְאָחִיד בֵּיהּ.
Come and behold: whoever deals in the Torah is attached to the Tree of Life and, since he is attached to the Tree of Life, he is part of everything, as it is written: "she is a tree of life to those who lay hold on her..." (Mishlei 3:18). 2. Rabbi Yitzchak said: Whoever deals in the Torah is free from everything, even free from death, as mentioned, since liberty, WHICH IS BINAH, dwells upon him and is attached to him.
What does this statement mean? The followig source texts give some detail on how Chazal understand this concept
take refuge under Your protecting wings. Selah.
וְרִבִּי יוֹחָנָן מַאי כוּלֵּי הַאי. דְּבָעֵי דְּיֵימְרוּן שְׁמַעְתָּא מִשְּׁמֵיהּ. דְּאַף דָּוִד בִּיקֵּשׁ עָלֶיהָ רַחֲמִים. שֶׁנֶּאֱמַר אָג֣וּרָה בְ֭אָהָלְךָ עוֹלָמִית אֶֽ֘חֱ֤סֶה בְסֵתֶ֭ר כְּנָפֶ֣יךָ סֶּֽלָה׃ וְכִי עָֽלְתָה עַל דַּעְתוֹ שֶׁל דָּוִד שֶׁיְּהֵא חַי וְקַייָם לְעוֹלָמִים. אֶלָּא כָךְ אָמַר דָּוִד לִפְנֵי הַקָּדוֹשׁ בָּרוּךְ הוּא. רִבּוֹנוֹ שֶׁל עוֹלָם. אֶזְכֶּה שֶׁיְּהוּ דְבָרַיי נֶאֱמָרִין בְּבָתֵּי כְנֶסִיּוֹת וּבְבָתֵּי מִדְרָשׁוֹת. שִׁמְעוֹן בֶּן נְזִירָא בְשֵׁם רִבִּי יִצְחָק אָמַר. כְּלֽ־תַּלְמִיד חָכָם שֶׁאוֹמְרִים דְּבַר הֲלָכָה מִפִּיו בָעוֹלָם הַזֶּה שְׂפָתָיו רוֹחֲשׁוֹת עִמּוֹ בַקֶּבֶר. שֶׁנֶּאֱמַר דּוֹבֵב֭ שִׂפְתֵ֥י יְשֵׁנִֽים. מַה כֹמֶר שֶׁל עֲנָבִים זֶה כֵּיוָן שֶׁמַּנִּיחַ אָדָם אֶצְבָּעוֹ עָלָיו מִיַּד דוֹבֵב אַף שִׂפתּוֹתֵיהֶם שֶׁל צַדִּיקִים כֵּיוָן שֶׁאוֹמְרִין דְּבַר הֲלָכָה מִפִּיהֶם שֶׁל צַדִּיקִים שִׂפְתּוֹתֵיהֶן מְרַחֲשׁוֹת עִמָּהֶן בַקֶּבֶר. מַה הֲנְָאָה לוֹ. בַּר נְזִירָא אָמַר. כְּבָדֵין דְּשָׁאתֵי קוֹנְדִּיטוֹן. רִבִּי יִצְחָק אָמַר. כְּבָדֵין דְּשָׁאת חֲמַר עָתִיק. אַף עַל גַּב דְּלֵית לֵיהּ טַעֲמָא.
Why did Rebbi Joḥanan insist to require that traditions should be reported in his name? For also David begged for divine mercy in this respect, as it says, may I dwell in Your tent forever, I shall seek shelter under the cover of Your wings, Selah. Could David think of living forever? Rather, David said before the Holy One, praise to Him: Master of the World, may I have the merit that my words will be mentioned in synagogues and houses of study. Simeon, the son of the Nazir said in the name of Rebbi Isaac If a Sage’s pronouncement in matters of practice is mentioned in this world, the latter’s lips whisper with him in the grave for it is said, dripping from the lips of the sleeping ones. Like that softened bunch of grapes, if a person puts his finger on it immediately it drips, so also the lips of the just drip, for if one mentions their pronouncements in matters of practice the latter’s lips whisper in the grave. What good does that do to them? The son of the Nazir says, like one who drinks spiced wine; Rebbi Isaac said, like one who drinks old wine, even though he cannot taste it.
When we speak at the seder, the widsom of our ancestors finds expression through us. When we learn from a haggadah, it is like they are teaching us directly. We keep the ideas of our ancestors alive.
"למען הביא ה' על אברהם וגו'" תני ר"ש בן יוחאי אומר כל מי שיש לו בן יגע בתורה כאילו לא מת שנאמר למען הביא ה' על אברהם את אשר דבר אליו לא נאמר אלא את אשר דבר ה' עליו:
"To bring Hashem to Abraham" Rabbi Shimon ben Yochai said: Anyone who has a son involved in Torah as if he did not die. As it is said "To bring Hasham on Abraham the things that were said about him" It is not written except "that which Hashem spoke about him"
דְּתַנְיָא אָמַר רַבִּי שִׁמְעוֹן, מֹשֶׁה לָא מִית. וְאִי תֵּימָא הָא כְּתִיב (דברים לא) וַיָּמָת שָׁם מֹשֶׁה. כַּךְ בְּכָל אֲתָר לְצַדִּיקַיָּא קָרֵי בְּהוּ מִיתָה. מַאי מִיתָה. מִסִּטְרָא דִּילָן אִקְרֵי הָכִי. דְּתַנְיָא אָמַר רַבִּי שִׁמְעוֹן, וְכֵן תָּנָא, דְּמַאן דְּאִיהוּ בִּשְׁלִימוּתָא דִּמְהֵימְנוּתָא קַדִּישָׁא תַּלְיָיא בֵּיהּ, לָא תַּלְיָיא בֵּיהּ מִיתָה וְלָא מִית. כְּמָה דְּהֲוָה בְּיַעֲקֹב דִּמְהֵימְנוּתָא שְׁלֵימָתָא הֲוָה בֵּיהּ.
888. We have learned that Rabbi Shimon said, Moses did not die. If you ask, it is written, "And Moses died there" (Devarim 34:5)! Also in every place where "death" is mentioned by the righteous ! What is death? From our view, it is called so. For we have learned that Rabbi Shimon said, someone who is in complete faith, holiness depends on him, and note death and he does not die. Just as Jacob had perfect faith.
וְאַתָּ֡ה אַל־תִּירָא֩ עַבְדִּ֨י יַעֲקֹ֤ב נְאֻם־יְהֹוָה֙ וְאַל־תֵּחַ֣ת יִשְׂרָאֵ֔ל כִּ֠י הִנְנִ֤י מוֹשִֽׁיעֲךָ֙ מֵֽרָח֔וֹק וְאֶֽת־זַרְעֲךָ֖ מֵאֶ֣רֶץ שִׁבְיָ֑ם וְשָׁ֧ב יַעֲקֹ֛ב וְשָׁקַ֥ט וְשַׁאֲנַ֖ן וְאֵ֥ין מַחֲרִֽיד׃
But you, Have no fear, My servant Jacob
—declares the LORD—
Be not dismayed, O Israel!
I will deliver you from far away,
Your folk from their land of captivity.
And Jacob shall again have calm
And quiet with none to trouble him;
רִבִּי יִצְחָק אָמַר, חַבְרַיָּיא הָא אוּקְמוּהָ, דִּכְתִּיב וְאֶת זַרְעֲךָ מֵאֶרֶץ שִׁבְיָם, מַה זַרְעוֹ בַּחַיִּים, אַף הוּא בַּחַיִּים.
Rabbi Yitzchak said, The friends have established it, as it is written, "And your seed from the land of their captivity" (Ibid. 10). Just as his seed is alive, so is he alive.
All of the rightious are considered as alive forever. Why? I suppose it is because of their affects on others. This is why Yaakov's "live" is dependant on his children. It is because they continue his mission. Taanit 5a also has this statement.
(צד"ק ג"ם גש"ם ק"ם סימן): שאלו מינין את רבן גמליאל מניין שהקדוש ברוך הוא מחיה מתים אמר להם מן התורה ומן הנביאים ומן הכתובים ולא קיבלו ממנו
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עד שאמר להם מקרא זה (דברים יא, כא) "אשר נשבע ה' לאבותיכם לתת להם" לכם לא נאמר, אלא להם מיכן לתחיית המתים מן התורה
Heretics asked Rabban Gamliel: From where that the Holy One, Blessed be He, revives the dead? He said to them from the Torah, from the Prophets, and from Writings, but they did not accept them.
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until he stated to them this verse: “That your days may be multiplied, and the days of your children, upon the land that the Lord took an oath to your forefathers to give them” (Deuteronomy 11:21). To you, is not stated; rather, but: “To them,” From here resurrection of the dead from the Torah.
We can see that the Torah views the fulfilment of promises to the forefathers through their descendants as if the promise is fulfilled directly with the forefathers. If we interpret the Torah literally, we have a proof for the ressurection. If we allow for metaphor, we are showing a deeper element of what the ressurection really means.