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I am asked to check whether or not $\{f_n\}_{n≥0}$ and $\{g_n\}_{n≥0}$ in $C([0,1], \mathbb{R}), \| . \|_{\infty}$ converges. Where $f_n(t) = t^n-t^{n+1}$ ang $g_n(t) = t^n - t^{2n}$

In order to do so, I simply check if $\| f_n(t) - f(t) \|_{\infty}$ converges and same for $g_n(t)$.

Now, what I don't understand is how we chose $f(t)$ and $g(t)$ I saw the solution nd they simply pick $f(t) = 0$ and $g(t) = 1/n$

Why were theose choices made?

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    What is that $1/n$ in the definition of $g$? – 2017-01-23
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    If there is a variable $n$ appearing in the definition of $g(t)$ in your solutions, then the solutions are wrong. – 2017-01-23
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    What is this $f(t)?$ – 2017-01-23

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Hint: Both $(f_n),(g_n)$ converge pointwise to $0$ on $[0,1].$ If $(f_n)$ converged uniformly to some $f\in C[0,1],$ then $f\equiv 0,$ since uniform convergence implies pointwise convergence. Same for $g.$ Does this happen?