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(1) Given the sequence of functions - defined on the interval [0,1], prove that: a) The sequence converges pointwise (to what limit ?) b) The sequence converges uniformly. (2) Given the sequence of functions fo(t) nx defined on the interval [0,1], prove that it converges pointwise and find the limit f(x). Is f continuous ? As a consequence, does the sequence fn converge uniformly to f ? (4) Prove that an = + n 1 n is decreasing and bounded below, thus convergent. The limit is called Euler's constant and is about 0.5772. Hints: for bounded below, use the Mean Value Theorein to get - and then add them all up. For decreasing, show that an+1 - an < 0.

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