Debemos luchar para alcanzar la grandeza.

English Translation

We must struggle to achieve greatness.

“luchar por” seems to be a lot more common: Google Ngram Viewer: luchar por,luchar para

Is “para” because it’s followed by an infinitive?

It’s not exactly the same meaning, luchar for X = fight for X, but luchar para Y = fight in order to Y. The phrase Y has an infinitive here because the subject is shared with the main phrase, but that’s not the trigger.

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English Translation

We must struggle to achieve greatness.

Se puede decir “esforzarnos” también?

If we simplify the sentence, leaving out the second infinitive, Ngrams only recognizes “luchar por la grandeza”.

I suppose you can.