てみせる details

write + (and…) + I'll show you I can — declares resolve by promising to prove it through the act itself

kaite miseru

I'll show you I can — declares resolve by promising to prove it through the act itself

meaning: prove I can write

今度こんどこそ、必かならず合格しごうかくしてみせる。

kondo koso, kanarazu goukakushite miseru.

This time for sure, I'll prove I can pass. neutral

見みてろよ、この壁かべ、絶対ぜったい登のぼってみせるから。

mitero yo, kono kabe, zettai nobotte miseru kara.

Just watch — I'm going to climb this wall, no matter what. casual

Examples

I'll show you I can — a declaration of resolve, proving the ability or determination by actually completing the act

今度こんどこそ、必かならず合格ごうかくしてみせる。

kondo koso, kanarazu goukakushitemiseru.

This time for sure, I'll prove I can pass. neutral

見みてろよ、この壁かべ、絶対ぜったい登のぼってみせるから。

mitero yo, kono kabe, zettai nobotte miseru kara.

Just watch — I'm going to climb this wall, no matter what. casual

一年いちねん以内いないに、必かならず結果けっかを出だしてみせます。

ichinen inai ni, kanarazu kekka o dashite misemasu.

Within a year, I will produce results — just watch. formal

A controllable act only — the speaker stakes their own effort on the
outcome, so no ×雨が降ってみせる. 必ず・絶対 often precede it, underlining the vow.

（よ）うと思う
Both voice a personal resolve about a future act, but ようと思う is private deliberation the speaker may still be weighing, while てみせる is a public promise staked on an audience — 合格してみせる commits in front of someone, 合格しようと思う merely reports a leaning.

てみる
One mora apart and built on the same 見る, but てみる tests an outcome ("try and see"), while てみせる stakes a claim on one ("I will, and I'll prove it") — 作ってみる is an experiment, 作ってみせる is a promise.