proof verification on the distance of a metric space

Question

let $A\subseteq \mathbb{R}$ and $\delta (A)=\sup\left \{ {| d(x,y)| :x,y\epsilon A} \right \}$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ cont. if $\delta (A)<+\infty $ then $\delta (f(A))<+\infty$

my solution,

we know that $\delta (A)=\delta (\bar{A})\Rightarrow \delta (\bar{A})<+\infty $ ,since $\bar{A}$ is closed and bounded we have that is compact ,now $f$ is cont. and we know that $f(\bar{A})$ is also bounded so $\delta (f(\bar{A}))<+\infty$ but

$\delta (f(A)) \subseteq \delta(f(\bar{A}))$(is = true??)

we have that $\delta (f(A))<+\infty$

Answer 1

It is true that $f(A) \subseteq f(\bar A)$; you should be able to convince yourself that in general, if $P \subseteq Q$, then $f(P) \subseteq f(Q)$.

From this, it follows that $\delta(f(A)) \leq \delta(f(\bar A))$.