Hi guys I just started metric spaces and was working on the following question. I need to prove by finite induction the following:-
Show that $x_1,\dots, x_n$ are n points in a metric space $(X,d)$ then
$$d(x_1,x_n) \leq d(x_1,x_2) +\dots+d(x_{n-1},x_n) $$
Now from based on the question by looking at a number of points for example 3 points
$$d(x_1,x_3) \leq d(x_1,x_2)+d(x_2,x_3)$$
therefore by adding a point we would then get the following
$$d(x_1,x_4) \leq d(x_1,x_2)+d(x_2,x_3)+d(x_3,x_4)$$
The following shows my induction proof but I am not familiar with proof by induction and was wondering if I have done it correctly if not can someone guide me:-
$$d(x_1,x_n) \leq d(x_1,x_n)$$ The above is my base case
Initial Hypothesis : Assume the following
$$d(x_1,x_k) \leq d(x_1,x_2)+\dots+d(x_{k-1},x_k)$$
Then the following holds:-
$$d(x_1,x_{k+1}) \leq d(x_1,x_2)+\dots+d(x_k,x_{k+1})$$
Best Answer
The base case is: $d(x_1,x_1)\leq d(x_1,x_1)$.
The inductive hypothesis is: Assume $d(x_1,x_k)\leq d(x_1,x_2)+\dots +d(x_{k-1},x_k)$ for some $k\geq 1$.
Then $$d(x_1,x_{k+1})\leq d(x_1,x_k)+d(x_k,x_{k+1})\leq d(x_1,x_2)+\dots +d(x_{k-1},x_k)+d(x_k,x_{k+1})$$
by the triangle inequality and the inductive hypothesis.