f_n does converge pointwise, and the pointwise limit of this function is indeed f(x) = 0.

In order to show that the limit of f_n(x) is zero, consider the following two cases:

x ≠ 0:

then, as you've stated, we can find an N such that whenever n>N, 2^-n and 2^(1-n) are both less than x (more precisely, we can choose N > 1 - log(x)/log(2) for any non-zero x). Since f_n(x) = 0 for infinitely many values, we conclude that f_n(x) converges (pointwise) to 0 for any non-zero x

x = 0:

For any n, 0 will fall into the interval 0<= x <2^(-n). It is thus clear that f_n(0) approaches 0 as n->∞.

I'm not sure what you meant with your statement "the limit f_n doesn't equal 0". Do you mean you have to prove that it doesn't converge uniformly?

# F_n(x) is a real-valued function defined on [0, 1] by the formula below. Prove that {f_n} converges pointwise.?

f_n= 0 if 0<= x <2^(-n)

= 2^(n/2) if 2^(-n)<= x <= 2^(1-n)

= 0 if 2^(1-n)< x <= 1

I'm supposed to prove that {f_n} converges pointwise to 0 but the limit f_n doesn't equal 0.

I get that when x is between 2^-n and 2^(1-n), the interval is shrinking and shifting to the left as n approaches infinity. My TA gave me a hint that I'm supposed to show that given any x, I can find N such that for n>N the interval will not include x. I still need more help with solving this problem, though...

= 2^(n/2) if 2^(-n)<= x <= 2^(1-n)

= 0 if 2^(1-n)< x <= 1

I'm supposed to prove that {f_n} converges pointwise to 0 but the limit f_n doesn't equal 0.

I get that when x is between 2^-n and 2^(1-n), the interval is shrinking and shifting to the left as n approaches infinity. My TA gave me a hint that I'm supposed to show that given any x, I can find N such that for n>N the interval will not include x. I still need more help with solving this problem, though...

## 1 Answers

### Trending

- Find cos(A+B) and the quadrant of angle (A+B), given cos(A)=1/root5, sin(B)=-4/5, where A is in quadrant 1, B is in quadrant 4.?
- Probability question?
- What is the core function of the integumentary system?
- Rectangle dimensions?
- How to find a side of a square ?
- (True or False) If I was born in 2000, the people who are a year older than me were born in the last year of the ‘90s?
- Why does math exist?
- How do you you do 800 divided by 12 in your head?
- What is the easiest way to work out this math problem? 3/7 of 5 is the same as 5/7 of 3. Is this statement true or false?
- Hi this is not a joke. my calculator is broken so can you help me figure out 9 plus 10. Thank you?