Verified answer

Proof by contradiction.

Suppose that t < 0.

Since {t(n)} converges to t, given any ε > 0, there exists a positive integer N such that

|t(n) - t| < ε for all n > N.

Choosing ε = (1/2) |t| yields a positive integer N such that |t(n) - t| < (1/2) |t| for all n > N.

==> (-1/2) |t| < t(n) - t < (1/2) |t| for all n > N.

==> t - (1/2) |t| < t(n) < t + (1/2) |t| for all n > N.

Since t < 0, we have that t + (1/2) |t| < 0.

Hence, t(n) < 0 for all n > N; this contradicts the hypothesis that t(n) ≥ 0 for all n.

Therefore, we conclude that t ≥ 0.

I hope this helps!