Let (X, 61) be a metric space and let S(x, n) and S(y, r2) be two intersecting balls containing a common point z. Show that there exists an r > 0 such that S(z, r%) C r1) n r2). Hint: Since z E S(x,...

Sol:
Given z is a point in S(x, r1)
We know that S(x, r1) is open.
Hence there exists an open ball S(z, r1’) centered at z and having a radius r1’ that is contained in S(x, r1).
Similarly, there exists S(z, r2’) that is centered a z and having r2’ and is contained in S(y, r2).i.e.
( ) ( )
( ) ( )
Let *
+
Then ( ) is subset of both S(x, r1) and S(x, r1)
Hence, we can say that ( ) ( ) ( )
Sol:
Let us consider a t in A.
If f is a limit point of the set, then there exists a sequence * + in the set such that the
limit uniformly. We know that, uniform convergence of a sequence implies point
wise convergence.
Hence we can write,


( ) ( )
Now, F is an intersection of such sets of C [0, 1], we can say that F is a closed subset of C [0, 1].
Hence we can say that,
* , - ( ) + , -
If you want more details on this question please refer to the theorem 2.1.24 (i) as mentioned in the
hint.
Sol:
The value of x lies in (0, 1+. Hence the value of 1/x lies in *1, ∞). The independent variable is
open on the left side and the dependent variable is open on...