Consider the sequence of functions {f} defined on the interval [0, 1], wherenxfn(z) =Vx € [0, 1], Vn € N.1+n²x²¹Show that the sequence of functions {f} is uniformly bounded, that is, there exists M >0 suchnxthat f(x)| ≤ M, Vx [0, 1], Vn N and evaluate lim 2 1n→∞1+n²x²1[0,1]

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Answered: Consider the sequence of functions {f}…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 30EQ

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Consider the sequence of functions {f} defined on the interval [0, 1], where fn(r)= nx 1+n²r²¹ Vx € [0, 1], \n € N. Show that the sequence of functions {f} is uniformly bounded, that is, there exists M > 0 such nx that |f₁(x)| ≤ M, \x[0, 1], \n € N and evaluate lim n→∞ 1+n²r²