This is what I've done:
My thoughts:
From trig => cos + sin = 1. So, is it something like
|coskx + sinkx| / |2^k| < or = (in particular = ) 1/2^k = M. Then since ΣM = Σ(1/2^k) converges (since 1/2^n approaches 0, even though it never attains it) Thus the given series converges uniformly and...
Please help me with this one:
Suppose f is a continuous real valued function on R - real #s and that 0<f(x)<3 for every x in R. Let F(x) = integral from 0 to x of f(t) dt.
a) Show that F is uniformly continuous on R
b) Suppose the continuity assumption is left out, but the function f is...
Consider the series
1+ Σ((1/(2^k))coskx + (1/(2^k))sinkx)
(a) Show that series converges for each x in R.
(b) Call the sum of the series f(x) and show that f is continuous on R = real numbers
My thoughts:
From trig => cos + sin = 1. So, is it something like
|coskx + sinkx| / |2^k| < or =...
Suppose f: R->R is continuous on all of R and B is bounded subset of R.
a) show cl(B) is bounded set
b) show image set f(B) must be bounded subset of R
c) suppose g:B->R is defined & continuous on B but not necessarily on all of R - real #s, Must g(B) be bounded subset of R? (Prove or give...