I am trying to prove that the perspective of a convex function is convex.
Problem Statement
I'm trying to prove that $f$ is convex
$$f(x,t) = tg(x/t)$$
where $g$ is a convex function, $x\in \mathbb{R}^n$, and $t>0$ is a positive scalar.
Proof
Since $g$ is convex, then $$g(\lambda x_1 + (1-\lambda)*y_1) \le \lambda g(x_1) + (1-\lambda)*t*g(y_1) \,\, \forall x_1, y_1$$
Let $x_1=x/t$ and let $y_1=y/t$, and multiply both sides by $t$:
$$t*g(\lambda x/t + (1-\lambda)*y/t) \le \lambda t * g(x/t) + (1-\lambda)*t*g(y/t)$$
Using definition of $f$ observe:
$$t*g(\lambda x/t + (1-\lambda)*y/t) = f(\lambda x + (1-\lambda)*y, t)$$
and
$$ \lambda t * g(x/t) + (1-\lambda)*t*g(y/t) = \lambda f(x,t) + (1-\lambda)*f(y,t)$$
Substituting those into the inequality:
$$f(\lambda x + (1-\lambda)*y, t) \le \lambda f(x,t) + (1-\lambda)*f(y,t)$$
which by definition means that $f$ is convex.
My proof doesn't use epigraphs, although almost every other proof I've seen does. E.g. this proof in number 6.
I want to know if my proof is acceptable without having used epigraphs? Thanks.