Topology - Composition of two isometric embeddings Prove that the composition of two isometric embeddings is an isometric embedding and 
that the composition of two isometries is an isometry.
I have been working on this problem for sometime, if anyone could please help me solve this problem as i can not even start it. Thanks
 A: You know that
$$
d_Y(f(x_1), f(x_2)) = d_X(x_1, x_2) \quad\forall x_1, x_2\in X
$$
$$
d_Z(g(y_1), g(y_2)) = d_Y(y_1, y_2) \quad\forall y_1, y_2 \in Y
$$
Now take $y_1 = f(x_1), y_2 = f(x_2) \in Y$; apply the second equation, and then the first. What happens?
A: Here's an intuitive explanation. (I won't quite tell you the answer, but pretty close.)
Suppose we have spaces $X,Y$ and $Z$ and isometric embeddings 
$$f : X \rightarrow Y, \; g : Y \rightarrow Z.$$
Now pick two points $x,x'$ in $X$. These points are a certain distance apart. Lets call it $\Delta$. If we map them over to $Y$ via $f$, then the resulting points $f(x)$ and $f(x')$ are still $\Delta$ distance apart (why?). If we map those points over again (to $Z$ this time, via $g$), then the resulting points $g(f(x))$ and $g(f(x'))$ are still $\Delta$ distance apart.
Conclusion. If $x$ and $x'$ are separated by distance $\Delta$, then so too are $(g \circ f)(x)$ and $(g \circ f)(x')$.
It should be easy to turn this intuitive explanation into a proof.
