Uniformly Bounded Functions: Proving Sequence Convergence

[chaotixmonjuish]

Prove that a sequence of uniformly convergent bounded functions is uniformly bounded.

Attempt at proof:

So first we observe the following: ||f_n||\leqM_n. Each function is bounded. Also, |f_n-f|\leq\epsilon for all n \geq N. First off, we observe that for finitely many f_n's, we have them bounded by M₁...M_N. Then for n\geq N M_N+1 acts as a bound.

So now we make a new M = max{M₁...M_N+1}.

I'm kind of stuck on how to demonstrate uniform boundedness with an inequality.

My second question is as follows:

Let X be a metric space with a metric d. Let a be a fixed point in X. Let p be a point in X. Define f_p=d(x,p)-d(x,a).

I want to show this is bounded.

For this one I have no idea.

 

[HallsofIvy]

The second one follows directly from the "triangle inequality": d(x,p)\le d(x,a)+ d(a,p)