～て見(み)せる

て見(み)せる, primarily written in hiragana, is an emphatic structure that indicates that the speaker is going to do (A) in a way that it is clear to the listener. This can be translated as 'I will definitely (A)', or 'I swear I will (A)', but the literal translation of 'I will show you that I'll do (A)' also makes the point clear.
To use てみせる, simply alter the verb preceding it into its て-form.

• お前(まえ)のために、次(つぎ)の試合(しあい)では勝(か)ってみせる！
  I will definitely win the next competition for you!
• 来週(らいしゅう)のテストで合格(ごうかく)してみせる！
  I will definitely pass the test next week!
• 足(あし)が痛(いた)いけど、完走(かんそう)してみせる！
  My legs hurt, but I am definitely going to finish the race!

Due to みせる requiring a target for who the speaker is showing, this grammar pattern will only be used when there is either directly someone that will be shown (A), or when the speaker is talking about showing someone (A) that does not happen to be there at the moment either physically or temporally. It will not be used when simply strongly stating that you will do (A), unless who sees that (A) is important. In a sentence like this, where the person being shown is not obvious, it will come across as 'I'll show everyone'.
As て見(み)せる literally just means to 'show (A)', it will often be used without any special emphasis. Whether it is being used emphatically or not will usually be obvious from the overall context of the sentence.