How to prove f is uniformly continuous

Show that, if f:[0, ∞) → R is continuous and has a finite limt at ∞, then f is uniformly continuous on [0, ∞).

Thank you.

Let ε > 0 be given.

Since f has a finite limit L at infinity, there exists a positive integer N such that
|f(x) - L| < ε/3 for all x ≥ N.

Next, since f is continuous on [0, N] which is compact, we conclude that
f is uniformly continuous on [0, N].

Hence, there exists δ > 0 such that for all x, y in [0, N] such that |x - y| < δ,
we have |f(x) - f(y)| < ε/3.

Now, let x, y be in [0, ∞) and |x - y| < δ.

(i) If x, y are in [0, N], then |f(x) - f(y)| < ε/3 < ε, as required.

(ii) If x, y > N, then |f(x) - f(y)| ≤ |f(x) - L| + |f(y) - L| < ε/3 + ε/3 < ε.

(iii) If x is in [0, N] and y > N, then
|f(x) - f(y)| ≤ |f(x) - f(N)| + |f(N) - f(y)| ≤ |f(x) - f(N)| + |f(N) - L| + |f(y) - L| < ε/3 + ε/3 + ε/3 = ε.

By (i)-(iii), f is uniformly continuous on [0, ∞).

I hope this helps!