Dirichlet proved that, for any real irrational number \(\xi\), there exist infinitely many rational numbers \(\frac{p}{q}\) such that \(|\xi-\frac{p}{q}|2\). Let \(\mathbf A_n, \;n>2\) denote the set of algebraic numbers of degree \(\leq n\). Let \(\alpha\in \mathbf A_n\) and \(H(\alpha)\) the height of \(\alpha\), that is the largest absolute value of the coefficients of its minimal polynomial. In 1961, \textit{E. Wirsing} [J. Reine Angew. Math. 206, 67--77 (1961; Zbl 0097.03503)] made the conjecture that, for any real number \(\xi\not\in \mathbf A_n\) and any real number \(\varepsilon>0\), there exist infinitely many algebraic numbers \(\alpha\in\mathbf A_n\) such that \[ |\xi-\alpha|\ll H(\alpha)^{-n-1-\varepsilon},\tag{1} \] where \(\ll\) is for the Vinogradov symbol and the implicit constant should depend on \(\xi,n\) and \(\varepsilon\) only. Later, \textit{W. M. Schmidt} [Diophantine Approximation. Berlin etc.: Springer (1980; Zbl 0421.10019)] conjectured the optimal exponent \(-n-1\) in (1). \textit{V. G. Sprindzhuk} [Izv. Akad. Nauk SSSR, Ser. Mat. 29, 379--436 (1965; Zbl 0156.05405)] showed that the conjecture of Wirsing is true at for almost all real numbers. In [loc. cit.] Wirsing proved that for any real number \(\xi\not\in \mathbf A_n\) there exist infinitely many algebraic numbers \(\alpha\in \mathbf A_n\) such that \[ |\xi-\alpha|\ll H(\alpha)^{-C(n)} \] where \(\lim_{n\to \infty}(C(n)-n/2)=2\) and the implicit constant depends only on \(\xi\) and \(n\). In 1993 V. I. Bernik and the author proved in [Dokl. Akad. Nauk Belarusi 37, No. 5, 9--11 (1993; Zbl 0811.11048)] that for any real number \(\xi\not\in \mathbf A_n\) there exist infinitely many algebraic numbers \(\alpha\in \mathbf A_n\) such that \[ |\xi-\alpha|\ll H(\alpha)^{-B(n)} \] where \(\lim_{n\to \infty}(B(n)-n/2)=3\) and the implicit constant depends only on \(\xi\) and \(n\). In this paper, the author improves all the previous estimates concerning the real case for \(n>2\) in the following theorem. Let \(n\) be an integer at least \(3\). Then for any real number \(\xi\not\in \mathbf A_n\) there exist infinitely many algebraic numbers \(\alpha\in\mathbf A_n\) such that \[ |\xi-\alpha|\ll H(\alpha)^{-A(n)}, \] where \(A(n)\) is the largest real root of the polynomial \[ \begin{multlined} T(x)=4x^5-(4n+18)x^4+(n^2+11n+30)x^3\\ -(2n^2+10n+22)x^2+(2n^2+7n+4)x+n^2-5n+2,\end{multlined} \] if \(n=3,4,5\) and \[ T(x)=2x^5-(n+12)x^4+(2n+30)x^3+(2n-41)x^2-(3n-29)x+2n-10 \] if \(n>5\). The implicit constant depends on \(\xi\) and \(n\) only. This table gives an example of comparison of \(C(n), B(n)\) and \(A(n)\) for some values of the integer \(n\). \[ \begin{matrix} n& C(n)& B(n)& A(n)\\ 3& 3.28&3.5&3.73\\ 4& 3.82& 4.12& 4.45\\ 5& 4.35&4.71&5.14\\ 10& 6.92&7.47& 8.06\\ 100& 51.99& 52.92& 53.84\\ \end{matrix} \]