In Shakespeare's "Sonnet 18," how does the poet prove that even death cannot boast about capturing the beauty of the fairer youth?

The heroic couplet is the summation of the argument presented in Sonnet 18 by Shakespeare; in its grammar and meaning it is complete:

So long as men can breathe, or eyes can see,/So long lives this [sonnet], and this gives life to thee.

Each quatrain of this early sonnet expresses an argument for the beauty of the fair youth.  In the first quatrain, for example, this beauty cannot be compared to something so temporal as "a summer's day" which is intemperate. Likewise, the "darling buds of May" are soon destroyed by "Rough Winds."

In the second quatrain, the poet reflects that beauty can fade from the climate changes, time, or "chance."  Simply put, Time alters beauty.  But, in the third quatrain, the poet argues that the beauty of the fair you will not fade

Nor lose possession of that fair thou owest;/Nor shall Death brag thou wander'st in his shade,

because the youth's beauty will be preserved in the "eternal lines" of the sonnet.  In these lines, the youth will "growest"; he will flourish in the minds and imaginations of all who read the sonnet, which "gives life to thee."  Clearly, the heroic couplet of Shakespeare's sonnet is a restatement of theme.